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UBC Theses and Dissertations

Some aspects of three and four-body dynamics Barkham, Peter George Douglas 1974

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SOME ASPECTS OF THREE AND FOUR-BODY DYNAMICS  by PETER G.D. BARKHAM 0.  B.Sc.  ( E n g . ) , Southampton U n i v e r s i t y ,  1967  M . A . S c , U n i v e r s i t y of B r i t i s h Columbia, •  "  1969  /  I  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering  We accept t h i s t h e s i s as conforming to required  standard  THE UNIVERSITY OF BRITISH COLUMBIA September,  1974  the  In presenting  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements  f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.  I f u r t h e r agree that permission f o r extensive coyping  of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his r e p r e s e n t a t i v e s . .  It i s understood that  p u b l i c a t i o n , i n part or i n whole, or the copying of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n  permission.  PETER G.D. BARKHAM  Department of  Electrical  Engineering  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date  W  •  117*.  ii  ABSTRACT  Two fundamental problems of c e l e s t i a l mechanics are considered: the s t e l l a r or planetary three-body problem and a r e l a t e d form of the r e s t r i c t e d four-body problem.  Although a number of c o n s t r a i n t s  are  imposed, no assumptions are made which could i n v a l i d a t e the f i n a l s o l u t i o n . A c o n s i s t e n t and r a t i o n a l approach to the a n a l y s i s of four-body systems has not p r e v i o u s l y been developed, and an attempt i s made here to describe problem e v o l u t i o n i n a systematic manner. In the p a r t i c u l a r three-body problem under c o n s i d e r a t i o n two masses, forming a close binary system, o r b i t a comparatively d i s t a n t mass.  A new l i t e r a l , p e r i o d i c s o l u t i o n of t h i s problem i s found i n  terms of a small parameter e, which i s r e l a t e d to the distance separating the binary system and the remaining mass, using the two v a r i a b l e expansion procedure. O(e^)  The s o l u t i o n i s accurate w i t h i n a constant  error  and uniformly v a l i d as e tends to zero f o r time i n t e r v a l s  0(e- ). 1 4  Two s p e c i f i c examples are chosen to v e r i f y the l i t e r a l s o l u t i o n , one of which r e l a t e s to the sun-earth-moon c o n f i g u r a t i o n of the s o l a r system.  The second example a p p l i e s to a problem of s t e l l a r motion  where the three masses are i n the r a t i o 20 : 1 : 1.  In both cases a  comparison of the a n a l y t i c a l s o l u t i o n with an equivalent n u m e r i c a l l y generated o r b i t shows .close agreement, with an e r r o r below 5 percent f o r the sun-earth-moon c o n f i g u r a t i o n and l e s s than 3 percent f o r the s t e l l a r system.  iii The four-body problem i s derived from the three-body case by introducing a p a r t i c l e of n e g l i g i b l e mass i n t o the close binary system.  Unique uniformly v a l i d s o l u t i o n s are found f o r motion near  both e q u i l a t e r a l t r i a n g l e points of the binary system i n terms of the small parameter e, where the primaries move i n accordance with the u n i f o r m l y - v a l i d three-body s o l u t i o n .  Accuracy, i n t h i s case, i s Q  maintained w i t h i n a constant e r r o r 0(e ), and the s o l u t i o n s are uniforml y v a l i d as e tends to zero f o r time i n t e r v a l s 0 ( e ~ ^ ) . p o s i t i o n errors near  and  Orbital  of the earth-moon system are found to  be less than 5 percent when numerically-generated p e r i o d i c s o l u t i o n s are used as a standard of comparison. The approach described here s h o u l d , i n g e n e r a l , be useful i n the a n a l y s i s of non-integrable dynamic systems, p a r t i c u l a r l y when i t i s f e a s i b l e to decompose the problem i n t o a number of s u b s i d i a r y cases.  iv  TABLE OF CONTENTS Page ABSTRACT  ii  TABLE OF CONTENTS  iv  LIST OF TABLES . . LIST OF FIGURES  . . . .  vii  .  ACKNOWLEDGEMENTS  viii . . . . . . . . . . . . . . . . . .  ix  Chapter 1.  2.  3.  INTRODUCTION  . 1 . 1  1.1  Background  1.1  1.2  Methodology  1.7  PERIPHERAL MATERIAL  2.1  2.1  Introduction  . . . . .  2.2  L i m i t a t i o n s and R e s t r i c t i o n s  2.3  N-body Dynamics  2.4  nomographic Solutions  2.5  Two-body Motion  2.6  N o n - i n e r t i a l Systems of Reference  2.7  Integrals i n Rotating Systems of Reference . . .  . .  • •  . . . . . . . . . .  . . . . . . . .  . . .  .  2.1 2.1  . . . . . .  2.8  . . .  2.14  . . . . . . .. . . . . . . . . . . . .  LIMIT PROCESS CONSIDERATIONS . . . . . . . . . . .  2.16 2.24 2.32  . . . .  3.1  . . . . . .  3.1  3.1  Introduction . . . . . . . . .  3.2  Reduction of the 3-body Problem  . . . . .  3.3  Reduction of the 4-body Problem  . . . .  . . .  3.3 3.10  V  Chapter  ' 3.4  Small-parameter  3.5  Exact Solutions  3.6  O r b i t a l Perturbations  3.7  4.  Expansions  3.13  .  3.15 . .  3.19  3.6.1  Perturbed 2-body motion  3.19  3.6.2  Perturbed 3-body motion .  3.21  The Two-variable Expansion Procedure 3.7.1  Definitions  3.7.2  M u l t i p l e s c a l e methods  THE THREE-BODY PROBLEM  3.26 .  3.28 3.34  .  4.1  4.1  Introduction  4.1  4.2  Preliminary A n a l y s i s . . . . . . . . . . . . . .  4.2  4.3  The Uniformly V a l i d S o l u t i o n . .  4.10  4.4  A Uniformly V a l i d S o l u t i o n of the Problem . •. . .  4.5 5.  Page  Restricted • •  4.4.1  H i l l ' s s i m p l i f i c a t i o n of the problem  4.4.2  H i l l ' s variation orbit  4.4.3  Further terms i n the s o l u t i o n  4.26  restricted . . . . . .  4.28  . . . . . . . . .  4.34 4.38  Discussion . .  4.44  THE FOUR-BODY PROBLEM,  5.1  5.1  Introduction  ^.1  5.2  The R e s t r i c t e d Three-body Problem  5.3  Four-body Motion Near L^ and L^  5.4  Uniformly V a l i d Solutions  5.5  Discussion.  . . . . . . .  5.2  . . . . . . . . . . .  5.9  . . .  . . . . . . . . . . . .  5.21 • • . • • •  5.42  vi Chapter 6.  Page SPECIFIC SOLUTIONS 6.1  Introduction  6.2  Three-body Orbits 6.2.1 6.2.2  7.  . .  6.1  . . . . . . .  6.1 .  6.1  P e r i o d i c s o l u t i o n s of y ( t ) = B(t)y + f ( t )  6.4  I t e r a t i v e method to determine a T-periodic orbit  6.7  6.2.3  P e r i o d i c earth o r b i t  6.9  6.2.4  P e r i o d i c o r b i t s f o r s t e l l a r motion . . .  6.16  6.3  Four-body Orbits Near L  6.4  Discussion  4  and Lg . . . . . . .  .  . . . .  6.29  CONCLUSION  REFERENCES APPENDICES  . . . . . . . . .  6.21  7.1 .. .'. . ' : .  . . . . . . .  . •  .  R1  vi i  LIST OF TABLES Table  3-1  p a  9  e  O r b i t a l Motion i n the n-body Problem f o r 2 ^ n < 4  3  -  2 6  viii  LIST OF FIGURES Figure  Page  2-1  I n i t i a l c o n f i g u r a t i o n f o r the n-body problem . . . .  2.9  2- 2  Primary c o n f i g u r a t i o n and coordinate systems . . . .  2.26  3- 1  Regions of motion f o r p and p^  3.4  3-2  Lagrange points of the r e s t r i c t e d problem  3- 3  Development of the four-body problem .  3.27  4- 1  Primary c o n f i g u r a t i o n f o r the r e s t r i c t e d problem . .  4.27  4-2  Primary c o n f i g u r a t i o n , with respect to the 5*, H* coordinate system, f o r the r e s t r i c t e d problem  4.30  6-1  P e r i o d i c earth o r b i t about £ = 0 . 0 1 2 1 5 0 ,  6.12  6-2  P o s i t i o n e r r o r s f o r the earth o r b i t over one period  6.14  Numerical and a n a l y t i c a l s o l u t i o n s about C =0.012150, n =0.0  6.15  6-3  2  2  6-4  • • •  2  . . . . .  n 0-0 • • .' 2  =  2  P e r i o d i c o r b i t f o r y - ^ l O , y = 0 . 5 , r,=50 about. £ = 0 , 5 , n =0.0 . . . . . . . . . . ! . . . . . . 9  2  2  6-5  P o s i t i o n errors f o r the s t e l l a r o r b i t over one period • • • • •  6-6  P e r i o d i c o r b i t s near L^ i n the four-body problem . .  6-7  3.18  . .  6.18  6.25  P e r i o d i c o r b i t s near L^ i n the four-body problem  .  . . . . . . . . . . . . .  6.26  6-8  P o s i t i o n errors f o r the o r b i t near L^ . . . . . . .  6.27  6-9  P o s i t i o n errors f o r the o r b i t near L5  6.28  • •  ix  ACKNOWLEDGEMENTS  I t i s a pleasure to acknowledge the advice and encouragement of my s u p e r v i s o r s , Dr. V . J . Modi and Dr. A . C . Soudack, during the i n c e p t i o n and presentation of t h i s work. The manuscript was typed by Mrs. M. E l l i s , and my wife Anne spent many hours preparing the mathematics f o r p r i n t i n g . This research was supported by Northern E l e c t r i c , the U n i v e r s i t y of B r i t i s h Columbia and the National Research Council of Canada under grants A-2181 and A-3138.  DEDICATION  To my parentsj whose generosity made this work  and  interest  possible.  1 T.l  1.  1.1  INTRODUCTION  Background The n-body problem of c e l e s t i a l  mechanics may be d e f i n e d ,  quite simply, as f o l l o w s : N p a r t i c l e s of a r b i t r a r y mass a t t r a c t each other according to the Newtonian law of g r a v i t a t i o n and are free to move i n space. I n i t i a l l y they move i n any given manner; determine t h e i r subsequent motion. No general s o l u t i o n of t h i s problem e x i s t s .  Only the two-body problem  i s considered s o l v e d , because properties of the t o t a l i t y of s o l u t i o n s are known, and even i n t h i s case the o r b i t a l  possible  coordinates  cannot be represented as e x p l i c i t closed-form functions of time. Methods which have been applied to i n v e s t i g a t e n-body motion can broadly be divided into three c a t e g o r i e s : and f o r m a l i s t i c dynamics.  qualitative,  quantitative  Q u a l i t a t i v e methods may be employed, f o r  example, to determine conditions f o r s t a b i l i t y , s o l u t i o n existence and bounded motion, but are not u s u a l l y h e l p f u l i n e s t a b l i s h i n g solutions.  specific  T h i s , i n g e n e r a l , i s the domain of q u a n t i t a t i v e or e x p e r i -  mental dynamics.  Again, however, we encounter a s i g n i f i c a n t l i m i t a t i o n ,  as the accumulated e f f e c t of q u a n t i z a t i o n e r r o r over large i n t e r v a l s of time may i n v a l i d a t e any r e s u l t s derived from a numerical process.  integration  Information about system behaviour along the e n t i r e  a x i s cannot therefore be obtained from a purely q u a n t i t a t i v e  time  approach.  1.2 P a r t i c u l a r s o l u t i o n s of a p e r i o d i c or asymptotic nature have, f o r t h i s reason, assumed a p o s i t i o n of central importance i n c e l e s t i a l  mechanics,  and they provide f o r m a l i s t i c dynamics with i t s nuclear s t r u c t u r e . The a n a l y s i s presented here i s concerned s p e c i f i c a l l y with p e r i o d i c o r b i t s i n the three and four-body problems.  No severe  r e s t r i c t i o n s are imposed on the p a r t i c i p a t i n g masses of the three-body problem, but i t i s assumed that two of the bodies form a close system i n o r b i t about the t h i r d .  This c o n f i g u r a t i o n w i l l be described as the  s t e l l a r three-body problem, although i t may apply e q u a l l y to g a l a c t i c or planetary systems.  The four-body case i s derived from the  stellar  three-body c o n f i g u r a t i o n by i n t r o d u c i n g an a d d i t i o n a l p a r t i c l e o f n e g l i g i b l e mass i n t o the close binary system.  There i s no d i f f i c u l t y  i n r e l a t i n g these models to our s o l a r system; the s t e l l a r problem, f o r example, i s a g e n e r a l i z a t i o n of the main problem of lunar theory, and the four-body model i s of considerable current i n t e r e s t i n s a t e l l i t e dynamics. The p e r i o d i c o r b i t s i n question emanate from homographic, or c o n f i g u r a t i o n - p r e s e r v i n g , s o l u t i o n s of the two and three-body problems.  A l l s o l u t i o n s of the two-body problem a r e , i n f a c t , homo-  g r a p h i c , but the p a r t i c u l a r example chosen here corresponds to c i r c u l a r o r b i t s i n an i n e r t i a l coordinate system.  A r o t a t i n g system o f axes i s  then selected i n which the two bodies appear to be s t a t i o n a r y .  The  three-body c o n f i g u r a t i o n i s a special case of Lagrange's homographic s o l u t i o n f o r the general three-body problem, where one of the masses i s n e g l i g i b l e and the remaining two move i n c i r c u l a r o r b i t s .  The  3 •  1.3  r e s u l t i n g s i t u a t i o n i s known as the r e s t r i c t e d problem.  Five homographic  t solutions exist  where the three bodies move i n coplanar o r b i t s ;  the  p o s i t i o n s of r e l a t i v e e q u i l i b r i u m f o r an i n f i n i t e s i m a l mass i n the r o t a t i n g coordinate system are c a l l e d Lagrange, l i b r a t i o n or e q u i l i b r i u m p o i n t s , and are u s u a l l y denoted by L-j, L£, •••» L ^ . points (L-,,  Three of these  and Lg) are located on the l i n e j o i n i n g the two p r i m a r i e s ,  while the remaining two form e q u i l a t e r a l t r i a n g l e s with the primary bodies at two of the v e r t i c e s . A survey of e x i s t i n g l i t e r a t u r e r e l a t e d to the r e s t r i c t e d problem can be found i n Szebehely's a u t h o r i t a t i v e t e x t •[!].  Major  reviews of the general three-body and n-body problems by Lovett [2] and several other authors appeared between 1896 and 1919 [ I ] .  Recent  1  work i s mentioned i n a. remarkable c o n t r i b u t i o n to Burrau's problem by Szebehely  [3]. Motion i n the v i c i n i t y of L^ and L ^ , at l e a s t f o r an i d e a l i z e d  earth-moon system, i s s t a b l e .  This s t a b i l i t y i s disturbed by the  g r a v i t a t i o n a l influence of a fourth body, but there s t i l l  remains the  p o s s i b i l i t y of p e r i o d i c or bounded motion i n these regions.  By analogy  with the three-body case, a four-body problem i n v o l v i n g one i n f i n i t e s imal mass i s known as the r e s t r i c t e d four-body problem, although no s p e c i f i c primary motion i s implied by t h i s d e s c r i p t i o n .  Celestial  mechanics entered a period of rapid development towards the c l o s e of the nineteenth century, f o l l o w i n g the fundamental researches of H i l l and Poincare, and i n t e r e s t i n the. four-body problem can be traced to These s o l u t i o n s are shown i n Figure 3-1 on page 3.18, q . v .  k 1.4 t h i s era;  Perhaps the f i r s t s i g n i f i c a n t i n v e s t i g a t i o n was by Moulton  [ 4 ] , who considered a r e s t r i c t e d four-body problem i n which the three primaries move according to Lagrange's c i r c u l a r homographic s o l u t i o n s o f . t h e general problem.  Twenty-eight homographic c o n f i g u r a t i o n s were  found a f t e r formulating an i n t e g r a l of motion f o r the i n f i n i t e s i m a l mass.  Moulton's r e s u l t s were extended by Lovett [5] to include p e r i o d i c  motion of the p a r t i c l e near these p o s i t i o n s of r e l a t i v e e q u i l i b r i u m . More r e c e n t l y Huang [6] proposed a model that has since been described as the v e r y - r e s t r i c t e d four-body problem, i n which the three primaries move i n c i r c u l a r coplanar o r b i t s .  The d e f i c i e n c i e s of Huang's  formulation are now widely accepted, but a s u b s t a n t i a l number of numerical and a n a l y t i c a l studies were based on the model.  very-restricted  The choice of primary motion therefore leads to a dichotomy  i n e x i s t i n g methods of a n a l y s i s , and those r e s u l t s which depend on Huang's model are grouped immediately below. DeVries [7] found a n a l y t i c a l s o l u t i o n s f o r a l i n e a r i z e d f o u r body problem near L ^ , and compared the predicted motion with numerical solutions.  Some a m p l i f i c a t i o n of these r e s u l t s appeared i n a review  of l i t e r a t u r e r e l a t i n g to Lagrange points by Steg and deVries Tapley et al. [9,10,11,12] used numerical methods to  [8].  investigate  motion i n the v i c i n i t y of L^ f o r a three-dimensional case i n which the earth-moon o r b i t a l plane i s i n c l i n e d at 5.15° to the e c l i p t i c . Cronin et. al. [13,14] considered three-dimensional motion of the f o u r t h p a r t i c l e , and derived-existence conditions f o r p e r i o d i c o r b i t s Lagrange p o i n t s .  near  In [15] Wolaver examined the e f f e c t of i n i t i a l  conditions on motion near L  A  and L  q  using l i n e a r i z e d equations.  5 1.5 Bernstein and E l l i s [16] derived l i n e a r i z e d equations of motion and applied Floquet theory to determine a necessary and s u f f i c i e n t c o n d i t i o n f o r bounded motion about L ^ .  Kolenkiewicz and Carpenter [17] used  numerical perturbation techniques to f i n d p e r i o d i c and a l m o s t - p e r i o d i c o r b i t s near L ^ .  A v a r i a t i o n of Huang's formulation has been considered by  Matas [18], who extended the r e s u l t s i n [14] to accommodate e l l i p t i c primary o r b i t s and the e f f e c t of s o l a r r a d i a t i o n pressure on s a t e l l i t e motion near earth-moon Lagrange p o i n t s .  The a n a l y s i s was f u r t h e r  modified [19] to include the e f f e c t of a r e s i s t i n g medium.  In [20]  the  same author investigated the motion of near-lunar s a t e l l i t e s when the primaries move i n e l l i p t i c o r b i t s and the influence of s o l a r r a d i a t i o n pressure i s taken i n t o account.  Luk'yanov [21] has considered  satellite  motion near the e q u i l a t e r a l t r i a n g l e points when the primary bodies form two n o n - i n t e r a c t i n g binary systems, obtaining r e s u l t s s i m i l a r to those given i n [ 8 ] .  The p o s i t i o n c o n t r o l of a s a t e l l i t e near a Lagrange  point has been investigated by Wolaver et al. [ 2 2 ] , who described the development of an optimal c o n t r o l l e r based on l i n e a r i z e d equations and the very r e s t r i c t e d primary model.  This system was tested f o r a r e a l i s t i c  model of primary motion, and provided e x c e l l e n t control over long periods of time. Reservations about v a l i d i t y of the v e r y - r e s t r i c t e d primary model were expressed by Szebehely [23] and a l s o Danby [ 2 4 ] , who proposed an a l t e r n a t i v e approach to account f o r secular p e r t u r b a t i o n s . The p r i n c i p a l d i f f i c u l t y i s to f i n d a s u i t a b l e d e s c r i p t i o n of primary motion, and a number of a n a l y t i c a l models have since been employed. Mohn and Kevorkian [25] presented the formulation of four-body equations  of motion corresponding to the lunar theories of de Pontecoulant and H i l l , with the i n t e n t i o n of determining asymptotic s e r i e s s o l u t i o n s near Lagrange p o i n t s . p t o t i c expansions primary model.  Shi and Eckstein [26] used-.-.^matched -asym-  to determine earth-moon t r a j e c t o r i e s f o r a r e a l i s t i c  These a n a l y t i c a l s o l u t i o n s were subsequently v e r i f i e d  by Kevorkian and Brachet [27].  In [28] Schechter extended an e a r l i e r  nonlinear a n a l y s i s by Breakwell and P r i n g l e [29]  to i n v e s t i g a t e  three-  dimensional s t a b i l i t y of the sun-perturbed Lagrange points L^ and Lg. A stable e l l i p t i c o r b i t was found, s i m i l a r i n some respects to a numeri c a l s o l u t i o n determined by Kolenkiewicz and Carpenter [30] and published at the same time.  This numerical r e s u l t was l a t e r d u p l i c a t e d  a n a l y t i c a l l y by Kamel and Breakwell [31]. employed a d i f f e r e n t primary model,  Note however, that Schechter  so that exact agreement between  the two o r b i t s should not have been expected.  The a n a l y t i c a l  obtained by Luk'yanov [21] were extended i n [32]  solutions  to include the i n d i r e c t  e f f e c t of s o l a r g r a v i t a t i o n on motion of the earth-moon system. Nicholson [33] has examined behaviour near the col l i n e a r e q u i l i b r i u m points f o r an i n t e r a c t i n g primary model to p r e d i c t  station-keeping  requirements at the translunar and c i s l u n a r points L ^ a n d l^. and Franca [34] i n t o the  Giacaglia  have reported some progress i n a s i m i l a r i n v e s t i g a t i o n  evolution  of  e q u i l i b r i u m points of the  orbits  near  earth-moon  sun-perturbed system.  col l i n e a r  Giacaglia  [35]  a l s o proved an important q u a l i t a t i v e r e s u l t i n the r e s t r i c t e d body problem where three primary masses are located at the  has four-  vertices  of an e q u i l a t e r a l t r i a n g l e according to Lagrange's c i r c u l a r homographic configuration.  A r e g u l a r i z i n g transformation i s found f o r t h i s case,  and extended to an (n+1) +  cf.  page  6.22.  body problem i n which n equal masses move i n  7 1.7 a regular polygonal c o n f i g u r a t i o n .  The numerical a n a l y s i s of f o u r -  body motion by Tapley et al. [9,10,11,12] has been extended by Tapley and Schutz [36] to a r e a l i s t i c s i t u a t i o n which includes the i n d i r e c t solar e f f e c t .  Bounded motion i n a region about  was found to be  possible f o r a period exceeding 5000 days. Many of the r e s u l t s i n d i c a t e d here are discussed by Szebehely [l ] 2  i n what i s perhaps the most comprehensive survey of motion near  e q u i l i b r i u m points of the r e s t r i c t e d problem.  1.2  Methodology These diverse r e s u l t s do. not, u n f o r t u n a t e l y , a f f o r d any s i g n i f i -  cant i n s i g h t i n t o the general" behaviour of four-body systems.  No  coherent process of development has been adopted, which i s s u r p r i s i n g f o r a problem of such inherent complexity, and as a consequence no detectable patterns have emerged. This a n a l y s i s takes, as i t s p r i n c i p a l o b j e c t i v e ,  the d e s c r i p t i o n  and development of a systematic a n a l y t i c a l approach to the four-body problem.  Some of the concepts mentioned i n Section 1.1 are a m p l i f i e d  i n Chapter 2.  Much of t h i s material i s a v a i l a b l e i n standard t e x t s ,  but i t was f e l t that an ommission of these fundamental r e s u l t s would detract from the subsequent a n a l y s i s .  In Chapter 3 the p a r t i c u l a r  three and four-body problems under consideration are decomposed i n t o a number of s u b s i d i a r y cases.  The process of r e - c o n s t r u c t i o n  leads  to a new d e s c r i p t i o n of system s t r u c t u r e , and a r a t i o n a l approach to both the s t e l l a r three-body and r e s t r i c t e d four-body problems now becomes p o s s i b l e .  8 1.8 One p a r t i c u l a r d i f f i c u l t y , known i n c e l e s t i a l mechanics as the problem of small d i v i s o r s , i s frequently encountered when nonlinear systems are studied using q u a s i - l i n e a r methods.  The associated  resonance  phenomena correspond to system d i s i n t e g r a t i o n , but as t h i s r a r e l y happens i n p r a c t i c e the small d i v i s o r problem i n d i c a t e s a l i m i t a t i o n i n the a n a l y t i c a l approach.  M u l t i p l e scale perturbation methods have r e c e n t l y  been developed to overcome r e s t r i c t i o n s of t h i s nature, and a twov a r i a b l e expansion procedure i s applied i n the present a n a l y s i s to solve the nonlinear equations of three and four-body motion.  The d e r i v a t i o n  of a unique u n i f o r m l y - v a l i d s o l u t i o n of the s t e l l a r three-body problem i s described i n Chapter 4, and i n Chapter 5  r e l a t e d s o l u t i o n s of  the r e s t r i c t e d four-body problem are determined f o r motion i n the v i c i n i t y of  and Lg.  These l i t e r a l s o l u t i o n s are compared with  numerically-generated p e r i o d i c o r b i t s f o r two d i s t i n c t primary models, one corresponding to the sun-earth-moon c o n f i g u r a t i o n of our s o l a r system, the other associated with a problem of s t e l l a r motion.  9 2-  2.  2.1  1  PERIPHERAL MATERIAL  Introduction  '  Before we embark on a d e t a i l e d study of the three and four-body problems, some general perspective of c e l e s t i a l mechanics i s  desirable.  Six topics of p a r t i c u l a r importance are c o l l e c t e d here, which serve both to introduce notation and to i n d i c a t e some of the basic s t r u c t u r e the n-body problem.  of  A number of standard d e r i v a t i o n s are quoted i n  condensed form, p r i m a r i l y so that the present work may be s e l f - c o n t a i n e d ; o r i g i n a l sources a r e , i n t h i s case, i n d i c a t e d i n the t e x t . The p h i l o s o p h i c a l basis of Newtonian dynamics i s b r i e f l y considered i n Section Z.Z, witn the i n t e n t i o n O T exposing the assumptions i m p l i c i t i n n o n - r e l a t i v i s t i c mechanics.  This i s followed by an i n t r o d u c -  t i o n to some general features of the n-body problem, and by a d e r i v a t i o n of equations of motion with respect to n o n - i n e r t i a l axes.  The chapter  i s concluded with a short discussion of i n t e g r a l s i n r o t a t i n g systems of reference.  2.2  L i m i t a t i o n s and R e s t r i c t i o n s Phenomenon:  A thing that appears, or is perceived applied  chiefly  or observed;  to a fact or occurrence,  of which is in question.  the cause  [Oxford]  Throughout the subsequent a n a l y s i s we s h a l l be concerned with the motion of a s p e c i f i c number of planetary or s t e l l a r bodies w i t h i n  10 2.2 a p a r t i c u l a r region of the universe. however,' mathematical.  The e n t i r e representation  As such i t should possess complete i n t e r n a l con-  s i s t e n c y , but t h i s i s no guarantee that t h e o r e t i c a l with observable r e a l i t y .  is,  results w i l l  agree  The extent of any such agreement i s determined  by the mathematical d e s c r i p t i o n of physical phenomena, and a l l s i m p l i f y ing assumptions w i l l tion.  lead to imprecision i n the process of  representa-  We therefore s t a r t t h i s i n v e s t i g a t i o n at the beginning, by  examining those fundamental p r i n c i p l e s upon which the development i s to be constructed. Although better models of r e a l i t y now e x i s t , t h i s a n a l y s i s takes as i t s foundation the laws of Newtonian mechanics. Law  I.  These are  [37 ]: 1  Every body continues i n i t s s t a t e of r e s t , or of uniform motion i n a r i g h t l i n e , unless  it  i s compelled to change that s t a t e by forces impressed upon i t . Law  II.  The change of motion i s proportional to the motive force impressed; and i s made i n the d i r e c t i o n of the r i g h t l i n e i n which that force i s impressed.  Law I I I .  To every action there i s always opposed an equal r e a c t i o n :  o r , the mutual actions of two  bodies upon each other are always e q u a l , and d i r e c t e d to contrary p a r t s . In the d i s c u s s i o n below, these laws w i l l be referred to as the  law of  i n e r t i a , motion and e q u i l i b r i u m r e s p e c t i v e l y .  i f we  I t i s necessary,  are to avoid a lengthy d i v e r s i o n i n t o philosophy (and, to some e x t e n t , h i s t o r y ) to accept c e r t a i n phenomena as being commonly understood. concepts of r e a l i t y , p o i n t , s t r a i g h t (or r i g h t ) l i n e , plane, space,  The  11 2.3 time, distance and force w i l l  therefore not be questioned here.  . Within a l o c a l region of the universe the laws of Newtonian mechanics generally produce t h e o r e t i c a l  results in excellent  agreement  with o b s e r v a t i o n , but we should be aware at the outset of t h e i r implications.  The study of dynamics, or motion i n time and space,  i n i t i a l l y requires that the observer possess standards of time and distance measurement which do not a l t e r r e l a t i v e to his system of reference,  f o r otherwise observations made under i d e n t i c a l conditions  would be i n c o n s i s t e n t .  In order to locate a s p e c i f i c event i n time  and space, the observer must s e l e c t a s u i t a b l e time o r i g i n and reference coordinate system.  I f the system o f reference i s d i f f e r e n t from h i s  own, however, there i s no guarantee that an event which he observes appear i d e n t i c a l i n the other system Of reference.  will  This subtle con-  s i d e r a t i o n i s i n t i m a t e l y connected with the process of transforming an event i n one system of reference i n t o a corresponding event i n another system.  Such a transformation a f f e c t s not only the p o s i t i o n ,  but a l s o the time i n s t a n t at which the event occurred, and therefore the influence of both time and space on t h i s operation must be known. The two phenomena may be regarded e i t h e r as being independent or i n t e r dependent; Newton, who had no reason to suspect otherwise, assumed space and time to be independent.  The l o g i c a l consequence o f t h i s  assumption i s that time a f f e c t s a l l space e q u a l l y , a c o n d i t i o n which Newton defined as "absolute" time [ 3 7 ] . 2  The properties of space are i m p l i c i t i n the law of i n e r t i a . I f t h i s law i s to be v a l i d i n the observer's coordinate system, then that system must have c e r t a i n s p e c i a l p r o p e r t i e s ; i n p a r t i c u l a r the space  12 2.4  defined by the system of reference must be E u c l i d e a n , f o r otherwise r e c t i l i n e a r motion would be a meaningless concept.  Suppose we have two  systems of reference i n which the standards of time and distance measurement are i d e n t i c a l , one moving with respect to the other through Euclidean space.  I f time i s an absolute, unchanging phenomenon i n a l l systems  of reference, the mathematical process o f instantaneous t r a n s i t i o n between systems i s e n t i r e l y l e g i t i m a t e .  Two points can therefore be  transformed instantaneously from one system i n t o another without a f f e c t ing the length which they d e f i n e .  Any r e l a t i v e motion of the two systems  w i l l have no e f f e c t on the 'transformation, but only because the t r a n s i t i o n is instantaneous.  When the r e l a t i v e motion i s known, a sequence of  events observed i n one system can, by t h i s process, be transformed i n t o an equivalent sequence with respect to the other system of reference.  If  the r e l a t i v e motion i s i t s e l f uniform and r e c t i l i n e a r , a uniform v e l o c i t y i n one system w i l l i n the other system.  transform i n t o a ( d i f f e r e n t ) uniform v e l o c i t y  C o n s e q u e n t l y , . i f the law of i n e r t i a i s true i n the  observer's coordinate system, i t w i l l a l s o be true i n a l l systems which move with uniform v e l o c i t y r e l a t i v e to the observer.  Now unless the  observer's system of reference i s free from a c c e l e r a t i o n , d e v i a t i o n s from the law of i n e r t i a w i l l be observed.  This immediately i m p l i e s  t h a t , somewhere i n space, there e x i s t s a f i x e d reference coordinate system associated with an unchanging or "absolute" space.  A l l systems of  reference i n which the law of i n e r t i a i s v a l i d must consequently be moving at uniform v e l o c i t y through t h i s absolute Euclidean space. Newton's law of motion i n t r o d u c e s , i n a d d i t i o n to time and d i s t a n c e , the concepts of force and mass.  These two phenomena are  13 2.5  r e l a t e d by an a l t e r n a t i v e statement of the law of motion:  a force F  a c t i n g on a p a r t i c l e produces an a c c e l e r a t i o n d i r e c t l y proportional to that f o r c e .  The constant, of p r o p o r t i o n a l i t y determines the i n e r t i a !  resistance o r "mass" of the p a r t i c l e , which must be an invariant.phenomenon i f the law of motion i s to be v a l i d of reference.  w i t h i n the observer's  system  This d e f i n i t i o n of mass can be applied to determine a  f u r t h e r q u a n t i t y , impulse, using the law of motion.  We have, i n the  d i r e c t i o n of the force F:  F  and hence  = r*v  •Lfc  where pndv i s the change i n momentum, and the derived quantity J i s defined as impulse.  I f two p a r t i c l e s c o l l i d e they w i l l  exert  impulsive forces on each other which, by Newton's law o f e q u i l i b r i u m , w i l l be equal and i n opposite d i r e c t i o n s .  The sum of the two impulses  w i l l consequently be z e r o , the t o t a l change i n momentum w i l l also be z e r o , and therefore momentum i s conserved during the c o l l i s i o n . Now mass time and distance must be i n v a r i a n t phenomena i f Newton's law of motion i to be u n i v e r s a l l y v a l i d ,  for  otherwise  changes.in momentum would  depend on the observer's system of reference.  No such r e s t r i c t i o n s  required by the law of conservation of momentum, even though i t i s derived from the law of motion. r e l a t i y i s t i c dynamics.  Therein l i e s i t s s i g n i f i c a n c e i n  •  I f the law of i n e r t i a i s v a l i d i n the observer's system of reference i t w i l l be true i n a l l systems moving at uniform v e l o c i t y  are  Ill 2.6 r e l a t i v e to the observer.  This statement applies equally to the law of  motion, but only i f mass i s assumed to be an i n v a r i a n t phenomenon.  In  g e n e r a l , transformations which preserve Newton's laws of i n e r t i a ! motion are of the form  |*  = fL J  + S. t + ^  ,  (2.1)  where ft i s a constant r o t a t i o n m a t r i x , a i s a constant vector d e f i n i n g the uniform r e l a t i v e v e l o c i t y and £ i s a constant displacement  vector.  A d e r i v a t i o n of t h i s transform, as i t applies to Newton's law of g r a v i t a t i o n , i s given by Wintner [ 3 8 ] . 1  The underlying assumptions of Newtonian mechanics can be summarized as f o l l o w s : 1.  Time and space are independent phenomena.  2.  Space i s Euclidean and absolute i n the sense that a f i x e d i n e r t i a ! reference must e x i s t .  3.  Mass, time and distance are fundamental, i n v a r i a n t phenomena. These a b s t r a c t , but s i g n i f i c a n t , l i m i t a t i o n s influence the  analysis.  The remaining r e s t r i c t i o n s  entire  are not s t r i c t l y necessary, but  s i m p l i f y the problem structure without departing too f a r from r e a l i t y . (i)  The dominant mass of the system i s s u f f i c i e n t l y d i s t a n t that i t s motion i s unaffected by any i n d i v i d u a l motion of the remaining masses.  This assumption i s unnecessary  the study of 2-body motion.  For more than two bodies,  except i n c e r t a i n s p e c i a l i n s t a n c e s ,  little  progress  in  15 2.7 appears to be possible without i t . on pages 2.25 and 2.26 (ii)  (See also the d i s c u s s i o n  ).  Only g r a v i t a t i o n a l f i e l d s r e s u l t i n g from the masses under consideration a f f e c t the motion of these bodies.  The  e f f e c t s of r a d i a t i o n pressure, t i d a l i n t e r a c t i o n ,  gravita-  t i o n a l f i e l d s of d i s t a n t masses, l i b r a t i o n a l motion and a l l other complicating features w i l l (iii)  be  neglected,  A l l the masses under consideration are r i g i d , bodies whose concentric l a y e r s are homogeneous.  spherical Each  may then be treated as a point mass, and Newton's law o f g r a v i t a t i o n can be a p p l i e d d i r e c t l y to determine motion of the point masses.  I f the i n d i v i d u a l dimensions of the  bodies are s m a l l , compared with the distances them, the inaccuracies negligible.  separating  introduced by t h i s assumption are  This t o p i c i s covered i n d e t a i l by Danby  [39 ]. 1  These three i d e a l i z a t i o n s , together with the i m p l i c i t assumptions of Newtonian mechanics, are p o t e n t i a l i r r e g u l a r i t i e s i n the foundation upon which t h i s analysis r e s t s .  E i n s t e i n disproved Newton's basic assump-  t i o n concerning the independence of time and space with the theory of s p e c i a l r e l a t i v i t y ; general r e l a t i v i t y f u r t h e r disproved the existence of absolute space, and i n d i c a t e d that space-time Euclidean continuum. theoretical predictions.  i s not generally a  The success of Newtonian mechanics as a v i a b l e  basis r e s t s , - however, on the general accuracy of i t s Only at v e l o c i t i e s approaching that of l i g h t , or i n the  presence of intense g r a v i t a t i o n a l f i e l d s , i s i t p o s s i b l e to detect  16 2.8 s i g n i f i c a n t c o n t r a d i c t i o n between theory and observation.  Neither con-  s i d e r a t i o n i s l i k e l y to a f f e c t the present a n a l y s i s .  2.3  N-body Dynamics The n-body ( o r , more a c c u r a t e l y ,  n-particle)  problem i s  concerned  with the motion of n p a r t i c l e s of a r b i t r a r y mass which a t t r a c t each other according to Newton's law of g r a v i t a t i o n .  At some time i n s t a n t  their  p o s i t i o n and v e l o c i t y vectors are known, and the subsequent motion i s to be determined.  This problem has not been s o l v e d .  Indeed, the n-body  problem i s so r e s i s t a n t  to a n a l y s i s that a general  s o l u t i o n e x i s t s only  f o r the two-body case.  The b a r r i e r to progress, which has so f a r proved  insurmountable, i s that 6n constants of i n t e g r a t i o n are necessary f o r a cOiiifJicoc 5 u l u c i 0 u  a r e , nevertheless,  uu t  only  ceil i i a v e  oeeii  TOunu.  Cci " i u l i i  opeolf'iC  CuSco  amenable to a n a l y s i s ; of these, the homographic  s o l u t i o n s considered i n Section 2.4 form an important c l a s s . " c l a s s i c a l " integrals  are  fundamental  to  any  study  of  The these  special c o n f i g u r a t i o n s , and we therefore give a b r i e f d e r i v a t i o n of the ten known i n t e g r a l s i n t h i s  section.  Newton's law of g r a v i t a t i o n can be expressed i n the f o l l o w i n g form: There are observed phenomena between two bodies i n space which can be described by presuming that two bodies  attract  each other with a force d i r e c t l y proportional to the product of t h e i r masses and i n v e r s e l y proportional to the square of the distance separating them. [40  ]  Suppose now that n p a r t i c l e s move i n a region of absolute space i n accordance with t h i s law and consider the i n i t i a l c o n f i g u r a t i o n shown  17  Figure 2-1  I n i t i a l c o n f i g u r a t i o n f o r the n-body problem  i n Figure 2-1, where we assume there e x i s t s some i n e r t i a l X , Y , Z reference system a r b i t r a r i l y located i n absolute space. p a r t i c l e , then s a t i s f i e s the equation:  The motion of p ^ , the i ' t h  18 2.10 where r^. = r^ - r..,  i s the mass o f  constant of g r a v i t a t i o n .  and k denotes the Gaussian  An existence c o n d i t i o n f o r s o l u t i o n s of t h i s  equation, due to P a i n l e v e , may be found i n P o l l a r d [41. ]; a proof appears 1  i n Wintner [ 3 8 ] . 2  Let the vectors r . , r . be given at some time i n s t a n t t = 0 at i i 3  which a l l the distances r . . are p o s i t i v e ; these w i l l be c a l l e d the i n i t i a l data.  If r(t)  denotes the smallest o f the distances r . . at time  t , then there e x i s t s a unique set of n vector functions r\(t) l a r g e s t i n t e r v a l of time - t  and a  < t < t-j containing the time i n s t a n t t = 0  2  such t h a t : (i)  r\(t)  s a t i s f i e s the d i f f e r e n t i a l Equation 2.2 f o r - t  (ii)  r\(t)  and r\(t)  / i i i \ \  4 I t y  -J - f  4-l-»V%  i •  V l l ^ .  agree with the i n i t i a l data when t = 0.  -Iri+A wi/^ 1 » i t <r<~ I  K U  . 4*  l  S,^  then r ( t ) -> 0 as t if t  2  < t < t-j;  2  y  4-  ^  S"  +• w  j  *?c  n n +  • ^  .• ^  +• l-»r» w  i wtr>v\»al  «i<>-  ' I.t v i  ^r*-i V  . «^ .  Also  4- <* -  t.  m  ^  y  t-j i f t-j i s f i n i t e and r ( t ) •+ 0 as t -> - t  is f i n i t e .  The essence of t h i s theorem i s contained i n i t s f i n a l  section,  which implies that a continuous s o l u t i o n e x i s t s during a time i n t e r v a l determined by the condition r(t) ->- 0.  We s h a l l consequently r e q u i r e ,  f o r the remainder of t h i s a n a l y s i s , that during the time i n t e r v a l of i n t e r e s t r{t) > r * > 0, where r * i s some f i x e d , p o s i t i v e lower bound on r(t). Notice that great care i s taken to avoid s t a t i n g , that a c o l l i s ion occurs when r -* 0.  r(t)  =  The d i f f i c u l t y i s  Min { r  1 2  , r^,  ••— ,  that  ) n  }  may tend to zero when none of the i n d i v i d u a l distances tends to z e r o ,  2  19 2.11 the r o l e of l e a s t distance being exchanged between them i n f i n i t e l y o f t e n . To prove that the condition  r{t) -> 0  corresponds to a c o l l i s i o n at  some f i n i t e time i n s t a n t t * i t i s necessary to show that the i n d i v i d u a l distances n > 3.  tend to l i m i t s as t -> t * ,  and t h i s has not been proved f o r  A discussion of t h i s enigma can be found i n Wintner [38 ] . 3  The c l a s s i c a l of motion.  i n t e g r a l s can r e a d i l y be obtained from the  equations  Performing the sum over i we o b t a i n :  1=1  The double sum over i and j i s e v i d e n t l y z e r o , as r\ . + r.^ = 0, and therefore fV \  -1  Z  *  :  F,  =  O  .  i  (,r. I  This can at once be integrated to give — ~ \  t  i  l-1  where a and b are constant v e c t o r s .  The time i n t e r v a l - t  d e f i n i n g s o l u t i o n existence w i l l n o t , i n g e n e r a l , p o i n t , but should be considered  1  I S. I  < t < t-j  y  be r e - s t a t e d a f t e r t h i s  i m p l i c i t i n the a n a l y s i s .  defines the centre of mass of the n p a r t i c l e s :  2  Now i f r  20 2.12 From Equations 2.5 and 2.6 we obtain the c o n d i t i o n f o r conservation of l i n e a r momentum:  r-  \  i* •  =  a. t + t  This r e s u l t , that the centre of mass moves with uniform v e l o c i t y i n the i n e r t i a ! system, i s to be expected i n the absence o f external  forces.  The i n e r t i a l reference system may now be located with i t s o r i g i n at the centre of mass of the n p a r t i c l e s without a f f e c t i n g Equation 2 . 3 . case a = 6 = 0, corresponding to the c o n d i t i o n r ( t )  = 0, and also  - t,  \ /  In t h i s  < t  ^ t,  .  With a and b d e f i n e d , 6 n - 6 constants of i n t e g r a t i o n remain to be determined. We now introduce a force f u n c t i o n F by the equation n.  (z.i)  If f = f ( r , , r ,  r ) , the operation - — i s defined by 3r„  9  5f  t  U f  ,  V . where a „ , 3,,, Yi/ e a r  system.  the components of  i n a Cartesian coordinate  Equation 2.2 may now be w r i t t e n i n the f o l l o w i n g form:  21 2.13  Taking the s c a l a r product with r\ and summing over i  «. s 1  and  therefore  Z  . C-I  cLt cLt  *  "*  Equation 2.11 can be i n t e g r a t e d , since r. . ?  i  1  d —  = ^ 7ft ( i * V ' r  t 0  g i v e  the energy i n t e g r a l  where S*, the t o t a l energy, i s an i n v a r i a n t q u a n t i t y .  The energy  S* reduces the number of undetermined i n t e g r a l s to 6n - 7-  constant  I f we now take  the vector product of r . with Equation 2 . 2 , and sum over i : -I  The double sum w i l l be z e r o , since r. x ?..•=  -r.. x ? , i  and therefore  22 2.14 Consequently IX,  where h * i s a constant vector d e f i n i n g the angular momentum.  The c o n d i t i o n  h . r = 0 defines a plane through the centre of mass c a l l e d the i n v a r i a b l e plane (provided h f  0).  the time i n t e r v a l - t  2  Suppose a plane IT contains the n p a r t i c l e s  < t < ty  during  The corresponding s o l u t i o n of Equation  i s then known as planar i f the p o s i t i o n of IT i n the i n e r t i a l system does not depend on t , and f l a t i f IT = i r ( t ) .  2.3  coordinate  If a solution is  p l a n a r , then TT i s the i n v a r i a b l e p l a n e , provided h * f 0; the proof appears in Wintner [ 3 8 " ] . The three i n t e g r a l s associated with h  complete the known i n t e g r a l s  of the n-body problem, leaving 6n-10 constants undetermined.  An exhaustive  treatment of t h i s e n t i r e subject may be found i n Wintner [38 ] . 5  l e s s d e t a i l e d a n a l y s i s , see P o l l a r d [41 ] 2  2.4  For a  and also Danby [ 3 9 ] . 2  Homographic Solutions A s o l u t i o n of the n-body problem i s c a l l e d homographic i f  the  c o n f i g u r a t i o n formed by the n p a r t i c l e s at a given time i n s t a n t t moves, with respect to the i n e r t i a l coordinate system, i n such a way that the c o n f i g u r a t i o n i s preserved as t v a r i e s .  If,  i n i t i a l l y , t = 0, then a  homographic s o l u t i o n w i l l be of the form  r-(b)  =  pCb)  H C O FCO) ,  L=l,2,r-,  23 2.15 where p(t) i s a s c a l a r ,  fi(t)  denotes a r o t a t i o n matrix and the i n e r t i a l  o r i g i n i s located at the centre of mass. (i)  There are two l i m i t i n g cases:  i f the c o n f i g u r a t i o n d i l a t e s without r o t a t i o n so that r (fc) L  = ^(fc") r ( o )  (  r \2 l  ,  i  the s o l u t i o n i s c a l l e d homothetic; (ii)  i f , conversely, the c o n f i g u r a t i o n i s . r o t a t i n g without dilation,  then  which defines a s o l u t i o n of r e l a t i v e e q u i l i b r i u m . Three r e s u l t s are of p a r t i c u l a r importance: (i) (ii) (iii)  a homographic s o l u t i o n which i s not f l a t must be homothetic; i f f l a t , a homographic s o l u t i o n must also be p l a n a r ; also a homographic s o l u t i o n i s a s o l u t i o n of r e l a t i v e e q u i l i b r i u m only i f i t i s planar and rotates w i t h a non-zero constant angular v e l o c i t y .  A proof of these statements may be found i n Wintner [ 3 8 L 6  S o l u t i o n s of  the two body problem are a l l homographic and, unless the angular momentum vector h * = 0, p l a n a r ; Lagrange's e q u i l i b r i u m s o l u t i o n s of the t h r e e body problem are both homographic and p l a n a r .  The s i g n i f i c a n c e  of  homographic s o l u t i o n s i s evident from these two cases, which together form the foundation of t h e o r e t i c a l  astronomy.  211 2.16 2.5  Two-body Motion. A s o l u t i o n of the problem of two bodies appears i n most texts  on c e l e s t i a l  mechanics.  Many r e s u l t s from the two-body problem w i l l ,  however, be needed l a t e r i n the a n a l y s i s ; f o r the sake of therefore,  completeness,  the elegant and concise development given i n Danby [39 ] 3  is  o u t l i n e d here. I f the i n e r t i a l o r i g i n i s located at the centre of mass of the two p a r t i c l e s ,  then s e t t i n g n = 2 i n Equation  r*,F, +  =  2.8:  o  (z. is)  We assume that neither mass i s i d e n t i c a l l y equal to zero. Equation 2 . 2 ,  and using Equation  2.14:  r  and t h e r e f o r e ,  if r = ?  f  where M =  2  Then, from  3  - r^  = - U H F a  + m . 2  Taking the vector product of f with Equation  (x. .?)  2.17:  25 2.17 and i n t e g r a t i n g :  (%'. 18")  r x r-  where h i s a constant v e c t o r , which we w i l l assume to be non-zero. A l t e r n a t i v e l y , i f Equation 2.13 i s used to determine the angular momentum:  The two momenta, although d i f f e r e n t i n magnitude, are e q u i v a l e n t .  Expand-  ing Equation 2.18 we o b t a i n :  1  1  '  which can be rearranged i n the form  1 +.. *s i-,*r,  r * r-  1 <-  !  s i n c e , from Equation  w , , ^ . The r e l a t i o n s h i p between h  2.14,  and h * i s then given by  M From Equation 2.18,  taking the s c a l a r product with r  26 2.18 so that the motion i s planar.  This i s the d i s t i n g u i s h i n g feature of the  two-body case; a s o l u t i o n i s possible because h i s normal to the i n v a r i a b l e plane.  Now take the vector product of h with Equation  using h x r = (r x r) x r from Equation 2.18.  2.17,  We obtain  = - k  =  - w^ H d. f  r  or  Integrating t h i s  expression  t *f  =  - tcMf  where P i s a constant vector.  -  ?  Because P i s i n the i n v a r i a b l e plane i t  f o l l o w s that P . h = 0, so one constant of i n t e g r a t i o n s t i l l be found.  (a.n )  remains.to  A parametric s o l u t i o n can, however, be found by e l i m i n a t i n g t  from Equation 2.19.  The s c a l a r product with r gives  27 2.19  But r x f = h ,  and therefore  I f the angle between P and r i s denoted by v , t h i s equation can be rearranged i n the form  u where  p  -p— = e ITM  ^ and  h -p— = p . ITM 2  This i s the equation of a conic with the o r i g i n at one focus; f o r an ellipse p = a(l-e  2  ), and f o r a hyperbola  p = a(e  2  - 1).  The vector  P points along the major axis of the o r b i t toward the p o s i t i o n of c l o s e s t approach between the two p a r t i c l e s ; v i s known as the true anomaly. Equation 2.18'may.be w r i t t e n i n the form  ptt  and from Equation 2.21 /  ( 1 •+ ecoso)  .  E l i m i n a t i n g r from Equation 2.22 we obtain  v  28 2.20 but, although t h i s equation can be i n t e g r a t e d , an a d d i t i o n a l s u b s t i t u t i o n i s necessary to put the r e s u l t i n a useful form. be integrated d i r e c t l y . d i f f i c u l t y ; when  Two cases can, however,  The c i r c u l a r o r b i t case (e = 0) presents no  e = 1 Equation 2.32 can be w r i t t e n  r =  sec- 0/  ,  z  and the corresponding form of Equation 2.24 becomes  It  which integrates to give 1*  T, the f i n a l constant of i n t e g r a t i o n f o r the p a r a b o l i c case, to the time i n s t a n t when  corresponds  v = 0.  Now consider the energy i n t e g r a l , which can be derived from a force function  F  =  We now w r i t e F  and consequently,  =  hV  •,  f o l l o w i n g the same procedure as i n Section  2.2:  29 2.21  ±  r .r  r  2-  where S i s an energy i n t e g r a l . from Equation 2.12  The corresponding expression derived  is (2.2.?)  r  where the two energy i n t e g r a l s S and S* are r e l a t e d by  n The constant S can.be evaluated from the c o n d i t i o n  r*. r  (2.2.S)  -  V  After substituting for ^  At. from Equation 2.22,  f o r e l l i p t i c o r b i t s we  obtain %$  «  U M r  0,0 -e*)  --.2  + r..  and, since r = 0 when r - a ( l ± e ) , Equation 2.26 may be w r i t t e n r  2. - _l_  r If Equation 2.18  i s squared:  (a-*<0  30  C  = (F.F)(F. r )  - (r.F)  (..*..3.o.)  which can be expressed i n the form  k  M  1.  r  r  t. . O.  — f r  where r . r i s obtained from Equation 2.29, and h  from Equation 2.22.  Now we define the e c c e n t r i c anomaly E by  r  = a- ( I - e cos E )  As the true anomaly n i f f o v o n t i a t i n n  v  Fnna-Hnn  varies from 0 to 2TT, E also varies from 0 to 2u. ?  3?:  which can be substituted into Equation 2 . 3 1 ; a f t e r some manipulation we obtain  = .1  The e c c e n t r i c anomaly i s chosen so that i t s d e r i v a t i v e i s p o s i t i v e , and therefore  (2.33)  31 2.23 Integrating t h i s expression over one complete o r b i t , the period r w i l l  ITT  be  C2.3O  PL  The mean motion n i s defined by  (2  .3<r)  P  and Equation 2.33 may be w r i t t e n  ait  = ( i - e<.c*E) crf-E  This can be integrated d i r e c t l y to give  n.C t - T )  =  £  - e,s^ E  where T, the time i n s t a n t at which r i s a minimum, i s the f i n a l constant of i n t e g r a t i o n f o r the e l l i p t i c o r b i t ; a s i m i l a r s o l u t i o n may be found f o r hyperbolic o r b i t s .  I f we replace M by m^ +  i n Equation 2.35,  the  mean motion i s determined by  C23?)  n.  The preceding a n a l y s i s i s based on the assumption that h f 0. I f , however, the angular momentum h i s z e r o , then from Equation 2.18 motion must be one-dimensional.  Although s t i l l homographic, t h i s  the one homothetic s o l u t i o n of the two-body problem.  An a n a l y s i s  is of  the  32 2.24 t h i s case, as i t applies to c o l l i s i o n o r b i t s and the process of r e g u l a r i z a t i o n , can be found i n Szebehely [ l  2.6  3  ].  N o n - i n e r t i a l Systems of Reference From t h i s point we s h a l l be concerned with the motion of one  p a r t i c l e under the g r a v i t a t i o n a l i n f l u e n c e of the remaining n-1 bodies. The equation of motion f o r t h i s p a r t i c l e w i t h respect to the i n e r t i a l coordinate system may be w r i t t e n i n the f o l l o w i n g dimensionless form  n. ~l a.t  Z  Z  .  lr-..|3  J ^  where p-j = k m.- ty z  3  normalization. ^pjr = r  , \ i s the distance normalization and T the time  We w i l l continue to use the convenient notation  to denote d i f f e r e n t i a t i o n with respect to normalized time,  t.  The four body problem under c o n s i d e r a t i o n i n t h i s a n a l y s i s subject to these c o n d i t i o n s : 1.  m > m,, > m » m^ ;  2.  the mass of p^ i s so small that i t has no e f f e c t on motion  1  3  of the primary bodies p ^ , p  2  and p  3  ;  We assume, i n t h i s f o r m u l a t i o n , that the mass m- of the p a r t i c l e i s not i d e n t i c a l l y equal to z e r o , i n which case the d i v i s i o n by m i n Equation 2.2 i s l e g i t i m a t e . i  is  33 2.25 3.  the i n e r t i a ! o r i g i n i s located at the centre of mass of the three p r i m a r i e s ;  4.  i f p* denotes a p a r t i c l e of mass  + m^ located at the centre  of mass of m^ and mg, then p^ and p* move i n two-body o r b i t s with respect to the i n e r t i a l system at an angular v e l o c i t y 'fi(t); 5.  with respect to the i n e r t i a l coordinate system, p^ and p^ move approximately i n two-body o r b i t s about t h e i r centre of mass;  6.  p^ moves i n the v i c i n i t y of p  2  and p^.  Conditions 2 and 3 are s t r i c t l y v a l i d only as m^ ->- 0; the f o u r t h assumption concerning motion of the centre of mass of p  2  and p-j needs some j u s t i f i -  cation. Tf v>* .  nncitinn  Ar>nr\-t-r\c r  - -  \iar- + nv  .  • _-  r>-f n r  with r o c n o r t tn r>*  , ... •  ,  then',- i n the notation of t h i s a n a l y s i s and with respect to the i n e r t i a l system of reference;  X*  where  <?F  P_ (co 5 ) = S  l\lco  S  x  S -11  i  and J  cos  S = F„ . r * —  [42 ]. 1  The f i r s t term of the force f u n c t i o n corresponds to the two-body s i t u a t i o n (see Equation 2 . 2 5 ) ; i f two body motion i s assumed, the r e l a t i v e magnitude p of the f i r s t neglected term i n the force f u n c t i o n i s given by  + n)  \  x  y  r*  )  The worst case r e s u l t s from m = m , and with the distance normalization 9  q  3k  chosen so that ^ 3 = 1 we obtain In p r a c t i c e p  f> £  ^  .  w i l l be s u f f i c i e n t l y small that deviations from two-body  motion can, f o r most purposes, be neglected.  35 2.27 The c o n f i g u r a t i o n of the three primaries i s shown i n Figure 2-2. S,H,Z  form an orthogonal coordinate system r o t a t i n g at a normalized  angular v e l o c i t y E,H  Q with respect to the i n e r t i a l  X , Y , Z system, w i t h  i n the o r b i t a l plane of the centre of mass of p  £,n,C to the  2  and p -  coordinate system rotates at an angular v e l o c i t y H,H,Z system, with  £,n  The  3  u> with  i n the o r b i t a l plane of p  2  respect  and p^.  In the i n e r t i a l system of reference n , the angular v e l o c i t y of the £,n,t; system, i s given by  Now suppose three p o s i t i o n vectors same point i n the X , Y , Z , tively.  E,H,Z and  £,n,C  a , 6 and y  define the  coordinate systems  respec-  Then:  where C i s the p o s i t i o n vector of B (the o r i g i n of the system) i n the r o t a t i n g  E,H,Z system.  Similarly:  £,n,C  coordinate  36 2.28  OC  (w + .n.) % y  +  [  C  ]  +  Z  "  a > <  I f r i s the p o s i t i o n vector of p  [ ']"HZ C  4  ^ ' ^- * ^  +  L  <  i n the £ , n , £  motion f o r p^ may be obtained from Equation 2.38  r  -*-  r  2-(l3 +st)x  11-  .i  + ux(unf)  L  3  where Equation 2.41  -f-  Z^i-K^^r)  a  +  C  *  c  system, the equation of as:  +• -^(xL* f) + ( i  r  **  J  i s used to express r i n the n o n - i n e r t i a l £ , n , S  system of reference. Transformations between the three coordinate systems -are determined by the f o l l o w i n g  relationships:.  A  X  cos <p  —  y'  o  cos y>  o  o  i  A  y A  z. •  and  o  0-43) A  2  37 2.29 (tos<jfcco£^ + S^H^Sin.^'toS'L)  ( - S i r t ( j £ u > S ^ +C o S ^ S i w ^ W u )  - S^^  SiwV  (2.4-4-)  A  A  COS V  z  where  fl = H z  ,  n. =  t  n£  and i denotes the angle between the  E , H and  The X , E and 5 axes coincide at the time o r i g i n  £, n  planes( Figure  t = 0.  2-2)  I f f i s defined  by  then with respect to the  n , C system of  reference:  (2.45)  A l s o , i f C = C E,  Z  -flay  t  t  then with respect to the E,  Sl^St^C.)  t  AxC  =[c.~  H , Z coordinate system:  JT- C-] £ a  +  [  -n- ] I c  (2-4-6)  38 2.30 Equations 2.43 and 2.44 impose no r e s t r i c t i o n s on s i n x, C, ft or n . Considerable s i m p l i f i c a t i o n i s , however, p o s s i b l e i f we assume: 1.  s i n i = 0;  2.  n = 0;  3.  the hypothetical body p* i s located at B, the o r i g i n of the £ , n» C system of  4.  reference;  p* and p-j move i n c i r c u l a r o r b i t s about the i n e r t i a l o r i g i n , i n which case C = C = fi = 0;  5.  the three primary bodies move i n coplanar o r b i t s , so that ^(t)  = 0 f o r i = 1, 2, 3.  Equation 2.42, which defines the motion of p  4  with respect to the £ , n , c  system of reference, can now be w r i t t e n  =  -si c  3  _  N  As a consequence of assumption 5 above, one p o s s i b l e s o l u t i o n Equation 2.49  is  39 2.31  £(t)  = £  (b)  = o  (a.jo)  which corresponds to motion i n the o r b i t a l plane of the primary bodies p.|, p  2  and p . 3  I t i s convenient, at t h i s p o i n t , to choose the time normalization so that n = 1.  I f we denote dimensioned q u a n t i t i e s by an a s t e r i s k ,  then  and consequently  x = 1/n*  gives the desired r e s u l t .  2.37 i s used to determine the mean distance  U tr x  PA  r  2 3  Now i f Equation  *:  (2.SZ)  j. +• rrt. 3 2-3  where  r  2 3  * = Ar  2 3  But we also have  and so Equation 2.52 may be w r i t t e n  n.  = 1 1 3  I f , t h e r e f o r e , the distance normalization A.is chosen so that X = then r  2 3  = 1 and also  r ^  ho 2.32  ^  -  + p-3  i  .  a  -^O  These normalized values are consistent with those i n Szebehely [ i * ] .  2.7  Integrals  i n Rotating Systems of Reference  I f a dynamical system i s described by the d i f f e r e n t i a l equation  and G(x, t ) i s a s c a l a r f u n c t i o n , then G i s an i n t e g r a l of Equation  2.55  if i  for a l l x , t  f fr(fc.t)l  O'  s a t i s f y i n g Equation 2.55.  body problem (Equation 2.12)  d(f , F , t )  =  =  j  \  2.  fr;  ll-Sb)  The energy i n t e g r a l of the n -  i s of t h i s form, where  *:r .F. c  - ...k N.  1  u,  i>c  l *jl r  In a r o t a t i n g coordinate system the equation of motion f o r a s i n g l e p a r t i c l e p^ of non-zero mass i s :  F,  + ^iixr.  •+  iix ( A x r J  + jflx F. -  (_VZL'  |r..p r 1  i  1  =  °  >  1*1 2.33 where ft i s the angular v e l o c i t y of the system with respect to an i n e r t i a l system of reference.  L  i  I f a force f u n c t i o n U i s defined as  Ir;.-  then Equation 2.57 may be w r i t t e n  .  >rc  .  and taking the s c a l a r product of t h i s equation with r\:  Now the vectors r . i n Equation 2.58  (which define the p o s i t i o n s of the  remaining n-1 p a r t i c l e s ) w i l l i n general be functions of time, so that  W- =  U. C r • , t )  and  Consequently, i f Equation 2.60 i s to s a t i s f y the c r i t e r i o n of Equation 2.56 f o r the existence of an i n t e g r a l , i t i s s u f f i c i e n t to require SI  =  ^Uat  =  O  ,  (.2.61 )  2.34 i n which case: 1.  the n o n - i n e r t i a l coordinate system rotates at a constant angular v e l o c i t y ;  2.  the n-1 p a r t i c l e s maintain a f i x e d c o n f i g u r a t i o n which i s unaffected by the motion of p^.  This l a s t c o n d i t i o n implies that the n-1 vectors r . (j + i ) define a J  homographic s o l u t i o n of the (n-1)  body problem, and from Section  2.4  t h i s s o l u t i o n of r e l a t i v e e q u i l i b r i u m must be p l a n a r . Subject to these r e s t r i c t i o n s , i f ; . ^  -  C"  1  p-j  +  the s c a l a r f u n c t i o n  r y . [ i x * (Cxi* r,.)  i s an i n t e g r a l of Equation 2.57,  =  constant  although t h i s expression i s  v a l i d only when m.. i s so small that c o n d i t i o n 2 above i s Both of these requirements  The corresponding  of  integral,  is  where the p o s i t i o n of p^ i s defined by  ( x , y , z ) , C i s the Jacobi constant,  ft i s normal to the i n v a r i a b l e x , y plane of p^ and p^, normalization i s chosen'so that ft = 1 . cussed i n Szebehely [ 1  fulfilled.  are s a t i s f i e d by the assumptions  the r e s t r i c t e d problem of three bodies. c a l l e d the Jacobi i n t e g r a l ,  strictly  and the time  The Jacobian i n t e g r a l  is dis-  ] , but note that Szebehely uses a modified force  k3 2.35 function  11"=  U. + > ^ p . ^ (  defined i n Wintner [38 ] 7  throughout his a n a l y s i s .  The i n t e g r a l  i s equivalent i n form to Equation 2.63  above.  Huang [ 6 ] has used " i n t e g r a l s " of motion to determine curves of zero v e l o c i t y i n a s p e c i f i c four-body problem. s a t i s f y Equation 2.56, consistent  The i n t e g r a l s do n o t ,  and the primary model f o r p.|, p  s o l u t i o n of the three-body problem.  2  however,  and p^ i s not a  Integrals of Equation  2.57 can be determined f o r the four-body case only i f Equation 2.61 valid.  is  This c o n d i t i o n i s s a t i s f i e d by Lagrange's homographic s o l u t i o n s  of r e l a t i v e e q u i l i b r i u m f o r the three primary masses, and corresponding surfaces of zero v e l o c i t y f o r the motion of a fourth p a r t i c l e i n f i n i t e s i m a l mass have been determined by Matas.  of  The f i r s t paper  presents surfaces of zero v e l o c i t y f o r the e q u i l a t e r a l  [43]  triangle configura-  t i o n and c e r t a i n s p e c i f i c values of primary mass; a l a t e r paper [44 extends these r e s u l t s to include the c o l l i n e a r c o n f i g u r a t i o n .  ]  Note  that surfaces of zero v e l o c i t y f o r the r e s t r i c t e d four-body problem were f i r s t determined by MouKon [ 4 ] , given.  although s p e c i f i c r e s u l t s were not  Matas, i n c i d e n t a l l y , does not mention Moulton's work.  This force function i s introduced i n [ 1 ] , Chapter 1, page 18, Equation 49.  3.1  3.  3.1  LIMIT PROCESS CONSIDERATIONS  Introduction For the remainder of t h i s a n a l y s i s we s h a l l concentrate on  the four-body problem described i n Section 2.5 and defined by Equations 2.47 and 2.48,  with c ( t ) = 0.  The motion of p.j, p  2  and p  ever, be known before attempting to solve these equations.  3  must, howFollowing  Huang's i n i t i a l statement of the " v e r y - r e s t r i c t e d four-body problem" [ 6  ]  i n 1960, many authors based t h e i r a n a l y s i s on the f o l l o w i n g two assumptions: (i)  p-j and p* move i n two-body o r b i t s with respect to the i n e r t i a l frame of reference;  (ii)  p  2  and p^ move i n two body o r b i t s about t h e i r centre of  mass. Although the r e s u l t i n g primary model i s p l e a s i n g l y uncomplicated, i t neglects the g r a v i t a t i o n a l i n t e r a c t i o n of p^ and p s o l u t i o n of the three-body problem.  2  and i s not a v a l i d  Discussions of t h i s d i f f i c u l t y  can be found i n Danby [24 ] and Szebehely [ 23, l ] . 5  More accurate  primary models, derived e i t h e r from numerical a n a l y s i s or lunar theory, have subsequently been employed to overcome t h i s l i m i t a t i o n [25 ] , [30 [31]  but unfortunately these precise s o l u t i o n s introduce i n t o the  equations of motion a v a r i e t y of parameters which tend to obscure  the  process of a n a l y s i s and made i t i n c r e a s i n g l y d i f f i c u l t to separate one perturbation from another.  ],  1|5  The four-body problem i s u s u a l l y treated as a perturbed version of the r e s t r i c t e d problem i n v o l v i n g p , p 2  3  and p^  so i t i s not unreason-  5  able to expect that Lagrange p o i n t s , which are of such s i g n i f i c a n c e  in  the r e s t r i c t e d problem, w i l l also be important i n the four-body case. In f a c t , with the exception of velocity surfaces,  two papers by Matas [ 43 , 44 ] on zero-  Shi and E c k s t e i n ' s  paper [ 2 6 ]  on Earth-Moon  t r a j e c t o r i e s and a r e l a t e d paper by Kevorkian and Brachet [27 ] , a l l recent work  on  the four-body problem i s concerned with motion near the  Lagrange points of p  and p .  2  3  A v a l i d s o l u t i o n of the four-body problem can only be obtained i f we f i r s t solve the three-body problem f o r p-j, p a corresponding s o l u t i o n f o r the motion of p^.  2  and p  3  and then derive  I n t u i t i v e l y i t seems  t h a t some "simplest" form of t h i s four-body problem must e x i s t , i n v o l v i n g o r b i t s of p  2  and p  probably  that are approximately c i r c u l a r and where  3  p^ moves near the Lagrange points of p  2  and p 3  Three questions  have  not, however, been r e s o l v e d . 1.  What i s the l i m i t process ( i f any) which reduces the threebody problem f o r p-j, p p  2.  2  2  and p  to a two-body problem i n v o l v i n g  and p ? 3  What l i m i t process reduces the four-body problem to a threebody problem i n v o l v i n g p , p 2  3.  3  3  and p^?  What i s e q u i v a l e n t , i n the four body case, to the Lagrange points of p  2  and p ? 3  The f i r s t two queries are answered i n Sections 3.2 and 3.3 below; the l a s t must wait u n t i l Chapter 5 f o r a d e f i n i t i v e answer, but the s o l u t i o n i s i n d i c a t e d i n Section  3.6.  3.3 3.2  Reduction of the 3-body Problem The p a r t i c u l a r three-body c o n f i g u r a t i o n which we s h a l l  consider  here i s determined by the c o n d i t i o n s : (i)  p* and p^ move i n c i r c u l a r o r b i t s about the i n e r t i a l origin;  (ii) (iii)  the three bodies p ^ , p^ and p^ move i n coplanar o r b i t s ; p  2  and p^ move approximately i n two-body e l l i p t i c o r b i t s  about  t h e i r centre of mass. The equation of motion f o r p^ w i t h respect to the i n e r t i a l frame of reference i s :  where we assume the mass of p  4  i s s u f f i c i e n t l y small to be neglected  (see c o n d i t i o n 2 on page 2 . 2 4 ) .  Condition ( i ) above implies t h a t  fi = C = C = 0, and we may a r b i t r a r i l y s e l e c t <L = 0. s i m p l i f i e s , and  Equation 2.42  i n the n o n - i n e r t i a l £ , n , £ coordinate system  3.1 takes the form:  which can be w r i t t e n as:  then  Equation  3.4  where the time normalization has been chosen so that n = 1. The c o n f i g u r a t i o n of p  2  and  must now be defined i n a manner  consistent with the assumption of (approximately) e l l i p t i c o r b i t s . general p  2  and p^ may be expected to move w i t h i n bounded regions of the  £ , n plane, and we f o l l o w Szebehely [ l so that £  9  In  > 0 (Figure  3-1).  6  J i n l o c a t i n g the l a r g e r mass m  2  14-8 3.5 I f the mean distance between p Equation 2.54)  u  3  = 1 - i ^ .  2  and p  i s chosen to be 1, then (from  3  Now the centre of mass of p  2  and p  3  is  permanently located at the o r i g i n of the £ , n axes, and consequently  X "  x  -  0 - / O  2 2 — = — m u m  since  y  3  .  From these two equations:  3  and we may therefore w r i t e  3 1  «I _L_  3  1  k9 3.6 The equations of motion f o r p  S ' t Z  1  +  5  +  H  +  =  nc z  2  now take the form  + |x,  tosC\-.o.)t  - g")  C3' -) 2  l r , z-3  ( 3 - I 3 )  1  where, f o r convenience, the s u b s c r i p t on £  2  ana  "  n  2  n a s  '  3 e e n  o m  itted.  two equations are s u f f i c i e n t to determine the motion of both p since £  2  a n  are known.  d ^3  c a n  be found from Equations 3.5 and 3.6 once £  2  2  These  and p^» and n  2  We can, moreover, r e a d i l y demonstrate that as the two  quantities  and  'Ml tend to z e r o , Equations 3.12 and 3.13 reduce to the equations of two-body motion i n a coordinate system r o t a t i n g with constant angular v e l o c i t y . the two-body case i n v o l v i n g p £,n,£  2  and p , the motion of p 3  2  In  i n the r o t a t i n g  coordinate system i s determined by  = o  ,  (1.1*")  5P 3.7 where n i s the (constant) angular v e l o c i t y with respect to an i n e r t i a l frame o f ' r e f e r e n c e .  This equation may be w r i t t e n  (I. ' O  jx  4  z  5^-  Ix  +  ( /  ix-*tj  where i t i s assumed n = I .  «  O  Equations 3.10 and 3.II  the present case, and so the motion of p  i  +  1  I  s  +  -  —  I  t J-  2  are s t i l l  i s defined by the  valid in  equations  1  -1  r  which are i d e n t i c a l with Equations 3.12 and 3.13 when the terms i n v o l v i n g 9  'C and u-| are set equal to zero. 2 We should therefore i n v e s t i g a t e the l i m i t process fi C -»- 0 and I f dimensioned q u a n t i t i e s are denoted by an I U asterisk,  3  then  and also r*  =  r, X  (3.2.0)  51 3.8  where x and A are the time and distance normalizations defined i n Section 2.6 (pages 2.24 and 2.31).  From the c o n d i t i o n that p* and p  1  move i n  c i r c u l a r o r b i t s about the i n e r t i a l o r i g i n we have, using Equation 2.37: k V  JTV-  where p*.  ( 3 - zi )  i s the normalized distance between p^ and the hypothetical p a r t i c l e The quantity C, which i s the normalized distance between p* and the  i n e r t i a l o r i g i n , i s determined by the c o n d i t i o n ,C  r, -  C )  =  (  wi  3  )  C  ,  so that  (3 . ZZ>  From Equations 3.21 and 3.22 we can now w r i t e  r  where the q u a n t i t y of p  2  and p .  e i t h e r m-j  3  k "^ / 1  1  3  C3.23)  i s completely determined by the c o n f i g u r a t i o n  I f Equation 3.23 tends to zero with T and X f i x e d , then  0 or r^  <=°.  The f i r s t c o n d i t i o n i s i n c o n s i s t e n t with the  52  3.9 +  requirement that m^ > the process r-j -> °°.  2 , and therefore ft C -> 0 must be equivalent  to  Now  and  r,. r,  X t-, . r  -  fc  r^.. F  +  t  ,  so that  1  r»\,  +  •L  0  —  (3.24)  which i s a form s i m i l a r to Equation 3.23.  The same argument can conse-  quently be a p p l i e d when Equation 3.24 tends to z e r o , provided the term -1  i s well-behaved as r-j -»• °°. bounded and 2 r  <  r  This c o n d i t i o n i s s a t i s f i e d when r  2  is  i » which "> c e r t a i n l y true i n the present s i t u a t i o n . s  Question 1 on page 3.2 problem f o r p-j, p  2  can now be answered as f o l l o w s :  the  and p , defined by Equations 3.12 and 3.13, 3  reduced to a two-body problem i n v o l v i n g p  2  the form e -*• 0, where z -»• 0 as r^ -»• » . ^Condition T on page 2.24  and p  three-body can be  by a l i m i t process of  3  We s h a l l , i n f a c t , take  states that m-j > m > m » 2  3  m^.  53 3.10 e = / 1/ \  3.3  1  ,  but t h i s choice w i l l be discussed l a t e r i n Section  3.4.  Reduction of the 4-body Problem In view of the s i m i l a r i t y between Equations 2.47 and 2.48  for  the four-body case and Equations 3.3 and 3.4 f o r three-body motion, i t might be a n t i c i p a t e d that the same l i m i t process can be a p p l i e d to both sets of equations.  To demonstrate the t r u t h of t h i s conjecture,  consider  the motion of p^ i n the r o t a t i n g £ , n , t, coordinate system under the g r a v i t a t i o n a l influence of p 3.14,  2  and Py  3.15 and 3.16 we may w r i t e :  ir**r A-  where £> n define the motion of p^. Equations 3.25 and 3.26 when the terms  and  By d i r e c t analogy with Equations  3  o  Equations 2.47 and 2.48 reduce to  5k tend to zero.  This condition i s i d e n t i c a l with that derived i n the pre-  ceding section f o r the reduction of Equations 3.12 and 3.13,  and conse-  quently the same l i m i t process can be applied i n the four-body case. This reduction of the three and four-body problems can be i n t e r p r e t e d i n the f o l l o w i n g way:  as p-j becomes more d i s t a n t , i t can  be assumed with increasing accuracy (i)  the motion of p  2  that  and p^ i s determined by two-body dynamics,  and (ii)  the motion of p , p 2  3  and p  4  i s determined by three-body  dynamics. This r e s u l t i s independent of the mass of p-j, provided i t  is  f i n i t e , since terms i n v o l v i n g m^ (through the q u a n t i t i e s ft C and  )  i n the equations of motion can always be rendered n e g l i g i b i l y small i f r is s u f f i c i e n t l y large.  In a more general a p p l i c a t i o n , t h i s d e r i v a t i o n  provides some j u s t i f i c a t i o n f o r neglecting the g r a v i t a t i o n a l e f f e c t of remotely d i s t a n t masses (see r e s t r i c t i o n ( i i ) on page 2 . 7 ) .  Suppose,  f o r example, that m and m^ correspond to the earth and moon mass 2  r e s p e c t i v e l y , and m^ i s the g a l a c t i c mass. so that Equation 2.52 may be w r i t t e n  rvt..  «-3  and, i f n = r ^ = 1, k "^ 7  we obtain =£:  i  In t h i s case m + 2  * m , 2  55 3.12 From Equations 2.43 and 2.44 the p o s i t i o n of p-j (with respect to the £ , . r i , £ system of reference) i s defined by  %  x  =  - r, cos  ( l - X O t  ;  =  r,  sc*.( t- X L )  t  I f these values are s u b s t i t u t e d i n t o Equations 3.12 and 3.13 a c a n c e l l a t i o n occurs (Appendix 1, Equations A1.4 and A 1 . 5 ) , and the leading term i n v o l v i n g m^ takes the form k^r*,  1  -  After substituting for  ....  k" t/ 1  1  we may w r i t e  3  l 17 For the case o u t l i n e d above (very approximately) —- = 0.4 x 10 and 11+ r = 8 x 10 , so that the c o e f f i c i e n t of the l a r g e s t d i s t u r b i n g term m  2  1  i s 0.8 x 1 0 " . 1 9  +  From A l l e n [45 ] : 20 Sun's distance from g a l a c t i c centre = 10 kpc G a l a c t i c mass  =  1.4 x 1 0  11  m  3.0857 x 10  o 6 We have — * 0.3 x 10 m m  m  T  2  17  — - 0.4 x 10  .  The distance normaliz-  2  ation A = 0.3844 x 10 m (page 4 . 3 7 ) , „so that r 9  m.  Q  where ni denotes the s o l a r mass. o (from page 4 . 3 7 ) , and therefore  =  1  - 8 x  10 . 11  56 3.13 3.4  Small-parameter  Expansions  We have now derived a reduction c o n d i t i o n , but not i n a useful form. and  The d i f f i c u l t y i s associated with the terms i n v o l v i n g  sin(l-ft)t  i n the equations of motion (2.47, 2.48,  3.12  cos(l-£2)t  and 3 . 1 3 ) ,  but i t can be resolved quite e l e g a n t l y by choosing a new time v a r i a b l e  t  = (\-Sh) t  4  .  ^ , =. k ^ t ^ m ,  From page 2.18  \  (3.2?)  , and from Equation 3.21  S1  L  r,»  3  so that i f we set (3-zs)  Equation 3.27 may be w r i t t e n t  +  =  [  I - e (n- j O * - ]  t  3  We s h a l l immediately drop the notation t , +  .  . (3.21)  and from t h i s point i n the  a n a l y s i s t , the independent v a r i a b l e , i s r e l a t e d to dimensioned time t * by f « 0 - . a ) f c * /  ' X  =  C l - J ^ ' t O t * , /  "17  = ( a * - J C l * ) t .  With respect to the new independent v a r i a b l e , Equations and  3.13 .take the form  (3.3a)  3.12  57 3.14  -1  I  (3.31)  =•  cot  r  +  ^Xti-S)  • . . ^ J_ -1  (5.32.)  where we have used the c o n d i t i o n and £ , n define the motion of p . 2  Equations 2.47 and 2.48 become  SL^C = u, /  t  from Equation  3.23,  S i m i l a r l y , f o r the four-body problem,  58 3.15 where, i n t h i s case, £ , n define the motion of p^ (the three and f o u r body cases w i l l be treated s e p a r a t e l y ,  so there i s no r i s k of c o n f u s i o n ) .  The expansions of  and  - ^, si^b  + |*, C 7, - «l)  are given i n Appendix 1.  Terms i n v o l v i n g (* / % c a n c e l , so that both {  2k expressions i n v o l v e sequences i n e  1  , where k = 3, 4, 5 ••• .  We have  consequently obtained equations of motion which depend on the small parameter e, where the l i m i t process e -> 0 reduces these equations to a form which, i n c e r t a i n cases, can be solved e x a c t l y .  These special  cases w i l l now be investigated i n more d e t a i l . 3.5  Exact Solutions When e = 0 Equations 3.31  and 3.32 reduce to Equations 3.17 and  3.18, which define the two-body motion of p 2  problem can be solved (Section 2 . 5 ,  Now although the two-body  Equations 2.32 and 2.36)  the s o l u t i o n  i s not an e x p l i c i t function of time and, even f o r t h i s comparatively simple case, expressions of the form £ = £ ( t ) , n = n ( t ) are d i f f i c u l t to obtain [ 42 ]. 2  An exact e x p l i c i t r e s u l t can, however, be determined f o r a  homographic s o l u t i o n of r e l a t i v e e q u i l i b r i u m .  In t h i s s p e c i f i c case a l l  d e r i v a t i v e terms are z e r o , and Equations 3.17 and 3.18 become  59 3.16 (3.  )  (336)  1  where the expression  J_  - 1  has been developed using Equation 3.9.  The p a r t i c l e p must therefore be located somewhere on the c i r c l e 2  S  % ^  1  =(1  -  ^ J  OS?)  1  For convenience we may take n = 0, so that  and, from Equations 3.5 and 3.6: (3.3<0  I f the p o s i t i o n s of p and p 2  3  are defined by Equations 3.38 and  3.39, then homographic e q u i l i b r i u m s o l u t i o n s can a d d i t i o n a l l y be determined f o r the three-body case derived from Equations 3.33 and 3.34 (defined by Equations 3.25 and 3.26). i n Equations 3.25 and 3.26:  S e t t i n g the d e r i v a t i v e terms equal to zero  60 3.17  = o ",1.  '<f3 i  = O  where 5, n denote the p o s i t i o n of p [l ], 7  we have substituted  Equation 3.41  is  n = 0,  r  y  3  = y,  4  and, to f o l l o w Szebehely's notation y  2  = 1 - y.  One s o l u t i o n of  i n which case  -  <>- .) s  :  Equation 3.40 then s i m p l i f i e s to ±  ft  (3.42)  = o  which defines the three c o l l i n e a r Lagrange points L-j, L The remaining s o l u t i o n of Equation 3.41  1  -  and L  3  [l ]. 8  is  = D  ti-h>  which, when 0 < y < 1,  2  (1.4-3)  reduces Equation 3.40 to the form  ( 1  '•V  3.4-0  61  3.18 1  -*—>  Figure 3-2  Lagrange points of the r e s t r i c t e d problem.  The r e s u l t i n g c o n d i t i o n ,  43'  =  C 3.4-5)  1  defines the two e q u i l a t e r a l t r i a n g l e points L^ and L^ (Figure  3-2).  An a n a l y s i s of the two and three-body equations of motion (Equations 3.17,  3.18,  3.25 and 3.26)  s o l u t i o n s i s now f e a s i b l e ,  i n the v i c i n i t y of these exact  but two a d d i t i o n a l small parameters,  a , must f i r s t be introduced.  6 and  The i m p l i c a t i o n s of t h i s process are  considered i n the f o l l o w i n g s e c t i o n .  62  3.19 3.6  Orbital  3.6.V  Perturbations  Perturbed 2-body motion The major obstacle preventing an e x p l i c i t s o l u t i o n of the  equations determining two and three-body motion i s the presence of nonl i n e a r terms i n v o l v i n g  _!  Close to an exact s o l u t i o n , however, these terms can be expanded as of a small o r b i t a l parameter.  For the two-body motion of p  2  functions  t h i s exact  s o l u t i o n i s given by Equation 3.38 and, i f we take  S  x  ( t )  =  I-p.,.  + l e f t )  ;  iJ-t}  a binomial expansion of the term  _J i r  form of Equations 3.17 and  [ e -2i  -3el  +  2e]  SfCb)  leads to a v a r i a t i o n a l x  3  l  3  - o  +  - 5  3ef  (3.46)  3.18:  L u\-fS>  S [ ?  -  (3.4*^  J  - o  (3.48 )  6  3 3.20  +  where 6 «  1.  Now suppose we consider the a d d i t i o n a l e f f e c t of p ^ , as  determined by Equations 3.31 and 3.32.  The motion of p^ w i l l depend both on  e and on 6, so that we should make the f o l l o w i n g m o d i f i c a t i o n to Equation  3.46:  where the v a r i a t i o n a l terms u ( t ) , v ( t ) correspond to the perturbation caused by p.j and e ( t ) ,  f(t)  determine the o r b i t a l p e r t u r b a t i o n .  The equations  of motion a r e , i n t h i s case, s i g n i f i c a n t l y more complicated, but as e ->• 0 with 6 f i x e d they reduce to Equations 3.47 and 3.48.  When 6 ->- 0  with e f i x e d we obtain  4-  To see t h i s , note that from Equation 3.9:  and t h e r e f o r e :  0 - f O * *"  _  S  Now take _ L _ 0-  HO  +V[e^f ]"?  Used-pi) 1  x  ...  <•  and expand (1 + r ) "  3 / z  .  ,  J  The expressions appearing i n Equations 3.47 and  3.48 f o l l o w d i r e c t l y from t h i s expansion.  6i+  3.21 U . | 1-26^(1+^,)^ 4-6^1 + /*,)^  - 2 v { l - 6  s  ( l Y , ^  £  -3,  <«ob + fx, ( g , - g ) x  3uV In., I  These equations describe the motion of p and 9  disturbance from r e l a t i v e e q u i l i b r i u m i s caused by p-|.  when the only Consequently  they define the " s i m p l e s t " v a l i d three-body model f o r p-j, p and p ^ , and 2  the s o l u t i o n of Equations 3.51 and 3.52 forms the basis of the " s i m p l e s t " four-body problem.  3.6.2  Perturbed 3-body motion The e x p l i c i t s o l u t i o n s f o r three-body motion are defined by  Equations 3.42 and 3.45.  I f the terms  i  and 3  are expanded about a p a r t i c u l a r Lagrange point  _j *3'  n » then w i t h L  we obtain the f o l l o w i n g v a r i a t i o n a l form of Equations 3.25 and 3.26 (see Section 5.2 and also Appendix I V ) :  65 3.22 Cr[  -  +  %y  where a « 1  0,  +/  j  +•  s r " 0  OC  and the c o e f f i c i e n t s v-j,  L . , f o r example,  o  {  =  -  3/.  ,  ^  =  >=  O  and 3£?  (3.5-4)  depend on  ( 1 - 2 . ^ )  and  and n ^ . °  3  ~  =  At •  4-  Equations 3.54 and 3.55 define motion about the Lagrange points i n the r e s t r i c t e d problem; a d e t a i l e d a n a l y s i s appears i n Szebehely [ i A more complicated s i t u a t i o n r e s u l t s i f p e l l i p t i c o r b i t s about t h e i r centre o f mass. motion of p  ?  ].  and p^ move i n  For small e c c e n t r i c i t y  can be described by Equation 3.46,  small parameter 6.  2  2  so that  the  _i  For the expansion of these nonlinear expressions we  require v a r i a b l e s , of the form  S^/t) =• $^ + -5"e^t:>+ t r s i ( b )  '  f  where the v a r i a t i o n a l terms e(t)  Y /tr) = Y  and f ( t )  a r i s i n g from the e l l i p t i c motion of p  2  u  +  3T^t> * .try f t }  ,  correspond to perturbations  and p^ (note that e(t)  and f ( t )  define v a r i a b l e s d i f f e r e n t from those i n Equations 3.49 and 3 . 5 0 ) . t  T  This d u p l i c a t i o n i s unfortunate, but these functions take no f u r t h e r part i n the a n a l y s i s and a l l non-ambiguous characters of the Greek and English alphabets have already been (or w i l l be) committed.  66 3.23 We-now have three possible s i t u a t i o n s i f the t r i v i a l case e -> 0,  6.-0  is excluded. 1.  When 6 -> 0 with a f i x e d the equations of motion reduce to Equations 3.54 and 3.55 f o r the r e s t r i c t e d problem.  2.  When a ->- 0 with 6 f i x e d , Equation 3.56 reduces to  . S^(t) - £ +• SeCt) u  ;  ^  = 7_ + S^Ct^ ,  O* ) 7  and consequently i t i s not p o s s i b l e to s p e c i f y an a r b i t r a r y i n i t i a l motion of p^ which i s independent of 6. 6 + 0, the functions 6e('t) and 6 f ( t )  A l s o , as  must tend to zero  uniformly i n t (see Section 3.7.1 f o r the d e f i n i t i o n of l i i l i i.Ci in i t,j'} •  iiiiS  COnuiClOii  c A c l UucS  icCti-lui"  s o l u t i o n and i s s u f f i c i e n t to determine e(t) uniquely.  tciinS  'fi'Ciii  tiic  and'f(t)  A proof of t h i s statement w i l l not be g i v e n , but  an analogous r e s u l t f o r four-body motion i s obtained i n Appendix V.  The o r b i t s defined by Equation 3.57 f o r the  e l l i p t i c r e s t r i c t e d problem correspond to the Lagrange e q u i l i b r i u m s o l u t i o n s of the ( c i r c u l a r ) r e s t r i c t e d problem. 3.  When both 6 and a are non-zero the r e s u l t i n g equations describe motion i n the e l l i p t i c r e s t r i c t e d problem where the i n i t i a l motion of p^ can be chosen a r b i t r a r i l y .  An- a n a l y s i s  of t h i s case, although with d i f f e r e n t forms of the equations of motion, can be found i n Szebehely [ I  9  ] and Nayfeh [ 4 6 ] . 1  I f we next take i n t o consideration the e f f e c t of py, the nonl i n e a r functions  _J  and  _!_  w i l l contain time-  67 3.24 varying terms depending a d d i t i o n a l l y on the small parameter e.  When  u ( t ) and v ( t ) correspond to the p e r t u r b a t i o n caused by p^ (note again that u(t) and v ( t ) define v a r i a b l e s d i f f e r e n t from those i n Equations 3.49 and 3.50)  then, with  +  (3.5-1?)  C3.S"0 e i g h t l i m i t - p r o c e s s combinations are p o s s i b l e .  Four of these i n v o l v e  the process  e •> 0, and consequently lead to s i t u a t i o n s described  previously.  The remaining combinations are l i s t e d below i n i n c r e a s i n g  order of complexity. (i)  6 and a ->• 0 with e f i x e d . p-j, p  2  and p  In t h i s case the motion of  i s defined by Equations 3.51  3  and 3.52, and  we obtain a s i t u a t i o n s i m i l a r to that described by Equation 3.57 where the i n i t i a l motion of p^ cannot be selected a r b i t r a r i l y .  Unique o r b i t s , which are the f o u r -  body equivalent of Lagrange e q u i l i b r i u m s o l u t i o n s of the r e s t r i c t e d problem, may be determined from the c o n d i t i o n that eu(t) and ev(t)  i n Equations 3.58 and 3.59 tend to  zero uniformly i n t as e ->- 0. (ii)  a  0 with <5 and e f i x e d .  case, where p orbits  2  In c o n t r a s t to the preceding  and p^ move i n approximately c i r c u l a r  ( i . e . e l l i p t i c o r b i t s with zero e c c e n t r i c i t y ) ,  c f . footnote on page 3. 22.  68  3.25 here the motion -of .p^ and p^ i s .approximately e l l i p t i c and  depends both on e and 6.  Apart'from t h i s complication  the same conditions a p p l y , and corresponding o r b i t s f o r p^ could be determined, (iii)  6 -> 0 with a and e f i x e d .  The i n i t i a l motion of p^ can now  be chosen a r b i t r a r i l y , and the r e s u l t i n g equations motion around the o r b i t s of case ( i ) (iv)  a , 6 and e are a l l non-zero.  define  above,  This f i n a l combination  describes the general s i t u a t i o n where p^ and p^ move i n approximately e l l i p t i c o r b i t s , and extends case ( i i i )  to  accommodate motion of p^ around the o r b i t s of case ( i i ) . To e l u c i d a t e the process of e v o l u t i o n described i n t h i s •  XI-, U l l l ^  j ~ M l / ^ I C ; . j . J I \ > l l  -C I I  _^ i l l V, I I  summarized i n Table 3-1.  T ^ I U  4-. — I  I, » « U  I "~  J.. „.~U4i„ v> t - i j r  An a s t e r i s k  \s t U  k n > l . . .v.4--J ,-n ~  4-„  I \**J  .  •  n . ^ ^ . v / i .  section, A ri ~i  in the e, 6 or a column denotes  a non-zero value of that parameter, and the comment " v a r i a t i o n a l o r b i t " in the f i n a l column denotes motion about the preceding  unique o r b i t f o r  an a r b i t r a r y i n i t i a l p o s i t i o n and v e l o c i t y of p^. The morphosis of t h i s 4-body problem i s i l l u s t r a t e d i n Figure 3-3.  Interactions  between the various p a r t i c l e s are represented by  f i g u r e s which correspond dimensionally to the number of bodies i n v o l v e d , with the exception of the enclosing four-dimensional hypersphere which i s depicted by a conventional sphere.  Nodes i n Figure 3-3 are equivalent  to the cases of Table 3-1, and the small parameter responsible f o r a t r a n s i t i o n from one node to another i s shown beside the corresponding branch.  The least-complicated s i t u a t i o n (node 1) i s associated with  69 3.26  Case  Motion of p  Motion of p^  and  e  6  1  0  0  Circular  2  0  *  El 1 i p t i c  3  0  Perturbed c i r c u l a r  4  * *  *  Perturbed e l l i p t i c  5  0  0  0  Circular  Lagrange point  6  -0  0  *  Circular  Variational  7  0  0  Elliptic  3-body unique o r b i t  8  0  * *  *  Elliptic  Variational  9  * * * *  0  0  Perturbed c i r c u l a r  4-body unique o r b i t  0  *  Perturbed c i r c u l a r  Variational  * *  0  Perturbed e l l i p t i c  4-body unique o r b i t  *  Perturbed el 1 i p t i c  Variational  10 11 12 Table 3-1  0  2  O r b i t a l motion i n the n-bodv Droblem f o r  orbit orbit orbit orbit  2*n^4.  e q u i l i b r i u m s o l u t i o n s of the r e s t r i c t e d problem (node 5) and with the simplest three-body model f o r p.|, p  2  and p^ (node 3).  form of the four-body problem, represented  The " s i m p l e s t "  by node 9, i s the l o g i c a l  consequence of these three fundamental cases.  A l l t h a t we now require  i s a method by which the n o n l i n e a r , non-homogeneous  equations of case 3  and case 9 can be s o l v e d .  3.7  The Two-variable Expansion Procedure Several perturbation methods have r e c e n t l y been developed to  determine approximate s o l u t i o n s of equations which, f o r a v a r i e t y of reasons, cannot be solved by conventional a n a l y s i s .  A review of these  71 3.28 perturbation techniques i s given by Nayfeh [46 ] , and also Cole [47 Certain concepts which r e l a t e to approximation methods i n general  ].  will  repeatedly be applied i n the present a n a l y s i s , arid f o r the sake of completeness these are defined here. Nayfeh [46  3.7.1  Further d e t a i l s can be found i n  ].  Definitions  Gauge f u n c t i o n s : I f f(e)  denotes a function which depends on the small  e, then, provided the l i m i t as e ->• 0 of f ( e )  parameter  e x i s t s , when e •*• 0  we could  have (i)  f(e)  (ii)  +0;  f(e) ->- A where  (iii)  f(e) -> °°  A is f i n i t e ;  (or f(e) -»-•-«>) .  The rate at which f(e) -*• 0 f(e)  or  f ( e ) -> °°  with a known gauge function g ( e ) ,  most useful  can be expressed by comparing  and of these the simplest and  are  e  e  The comparison of f(e)  ,  I  e ,  , e  j  •  ••  with g(e) employs the Landau symbols, 0 and 0 .  Large 0 I f there e x i s t s a p o s i t i v e number A independent of e and some e„ > 0 such that 0  72 3.29  I iU)  I  £= fl I gCe>l  V  |e|  ^  fe o  »  then we w r i t e  He)  =  0[  gCe)]  as  e -> O  The c o n d i t i o n f o r Equation 3.60 to be v a l i d can also be stated i n the form  e -5>e>  U)  5  I f f i s a f u n c t i o n of another v a r i a b l e x i n a d d i t i o n to e, and g ( x ,  e)  i s a corresponding gauge f u n c t i o n , then  as  €  —>  O  i f there e x i s t s a p o s i t i v e number A independent of e and an e  (3 .62.)  Q  > 0 such  that  When A and e  Q  are independent of x , then the c o n d i t i o n defined by  Equation 3.62 i s s a i d to hold u n i f o r m l y .  73 3.30 Small o: I f , f o r every p o s i t i v e number 6 independent of e, there e x i s t s an e  Q  such that  I He) I  £  l (e)| 3  for  U l  as  £  ±  6  then we w r i t e  He)  o [ g(e)J  =  —>  O  ( 3 . 6 3 )  The d e f i n i n g c o n d i t i o n can be replaced by £0)  =  (3. 64)  o  which i s a form corresponding to Equation 3.61  Asymptotic Expansions: A sequence o f functions <5 (e) i s c a l l e d an asymptotic sequence n  if  n  L  r»--i  J  as  e ->• 0  I f 5 (e) i s an asymptotic sequence and a m  sum  \  a  HI - O  f i f and. only i f  S  (e)  m  (3.65)  i s independent of e , the i n f i n i t e  i s the asymptotic expansion o f a function  71+  3.31  &  = 2^  ^S^Cfe}  as e + 0  ° L"  +"  .  (3.66)  We then w r i t e  Uniform v a l i d i t y : I f x i s a s c a l a r or vector v a r i a b l e independent of e, then the asymptotic expansion of f ( x ; e) i n terms of the asymptotic sequence nr A/  :  V  a  ( O T ^ e ^  as  e + 0  where the functions a (x) are independent of e. m  This expansion i s  uniformly v a l i d i f  / . — i  -o  where  uniformly f o r a l l x of i n t e r e s t . satisfied,  a  m  ( ) x  5  m  ( ) E  m u s t  b e  For these uniformity c o n d i t i o n s to be  small i n comparison to the preceding term  75  3. a _ ( x ) 6 _ ( e ) f o r each m. m  1  m  Now from Equation 3.65 6 (e) = o[6  1  ^ e ' ) ] so  t h a t , f o r the expansion to be uniform, a (x) must be no more s i n g u l a r m  than  _-|(>0-  a m  Each term i n the expansion must consequently be a small  c o r r e c t i o n to i t s predecessor,  independent of the value of x .  This  concept forms the foundation of the method of m u l t i p l e scales [ 4 6 ] , of 2  which the two-variable expansion procedure i s a s p e c i f i c case.  Elementary operations on asymptotic  expansions:  Consider the two asymptotic expansions  as  e -> 0  C3.7i)  as  e + 0 ,  C3.72)  and  where  <f> (e) i s an asymptotic sequence.  The operation o f l i n e a r combin-  m  a t i o n i s j u s t i f i e d i n g e n e r a l , and we have 00  •«K*jO +  Cx;0 ^  1 ^  s e  - ^  * P"  »  ^tS^  as e  0.  (3.73 )  I f , however, the two expansions are f o r m a l l y m u l t i p l i e d , the product 4>,-(e)<J>.(e) (where i , j = 0 , 1, 2 , •••• °°) cannot i n general be arranged in an asymptotic sequence.  The operation o f m u l t i p l i c a t i o n i s therefore  j u s t i f i e d only when the product 6. (e)<j>-(e) forms an asymptotic sequence or possesses an asymptotic expansion.  This c o n d i t i o n i s s a t i s f i e d f o r  the important case <j> (e) = e , and we obtain m  m  76 3.  6 (x^a)  3  C;x ;  e)  A / \  ( O  e  "  e -> 0  as  (3.74)  where  (3.7S)  K_^>  2_  I f f ( x ; e) and a (x) m  i n Equation 3.71  are i n t e g r a b l e  functions  of x , then  »c  as  e ->• 0  so that term by term i n t e g r a t i o n of an asymptotic expansion i s j u s t i f i e d . When f ( x ; e) and <i> (£). are integrable functions of e a corresponding m  r e s u l t holds f o r i n t e g r a t i o n with respect to e.  The process  of  d i f f e r e n t i a t i o n with respect to x o r e cannot, however, be j u s t i f i e d i n the general case.  We cannot, consequently, assume without r e s e r v a t i o n  that CO  as  e -> 0  (3.?0  S i m i l a r l y the process of exponentiation i s not j u s t i f i e d i n g e n e r a l , so that  2^  as  s •+ 0  (3.7f>  .  77  3.34 may not be v a l i d .  In both cases, however, when these operations are not  j u s t i f i e d they introduce non-uniformities i n t o the r e s u l t i n g expansions. The basic reference f o r a l l the r e s u l t s o u t l i n e d above i s Van der Corput's 1956 paper [ 4 8 ] .  A d d i t i o n a l discussions appear i n Nayfeh [46 ] 3  and  E r d e l y i [49 ] .  3.7.2  M u l t i p l e scale methods The two-variable expansion procedure introduced by Cole and  Kevorkian [50 ] , and developed by Kevorkian [51 ] , i s one of a c l a s s of methods i n v o l v i n g transformations of both dependent and independent variables.  A d e t a i l e d d e s c r i p t i o n of these m u l t i p l e scale asymptotic  methods can be found i n Nayfeh [46 ] , and only those aspects which apply to the present a n a l y s i s are discussed here. We are interested i n d e r i v i n g r e s u l t s which are v a l i d over long periods of time, and t h i s c o n d i t i o n w i l l be s a t i s f i e d i f the s o l u t i o n s are uniformly v a l i d i n t over some " l a r g e " time i n t e r v a l . Consider, f o r example, Equations 3.51 body motion of p  and p^.  2  and 3.52 which describe the  three-  The v a r i a b l e s u ( t ) and v ( t ) are assumed to  possess uniformly v a l i d asymptotic expansions i n terms of two t i m e - l i k e v a r i a b l e s x and r , so that oo u. (t^)  ro  \ L—j  or  L—i  u( m  t  v)  e  m  as  e 0  (3.?s)  78 3.35 where R ( x , T; e) = 0(e )  uniformly i n T and r as c + 0.  n  n  The formal  operation of d i f f e r e n t i a t i o n (as defined i n Equation 3.76)  i s assumed  to be l e g i t i m a t e , and from Equation 3.78 we obtain oo A/  m  V  6.  as  e  0  .  C3.8o)  m -o Corresponding expansions apply f o r v ( t ) and ^ f o r u ( T , r ) - i n Equations 3.78  to  when v ( x , r ) i s  substituted  m  3.80.  The t i m e - l i k e v a r i a b l e s T and r are selected so that one, r instance,  i s a n e a r - i d e n t i t y transformation of t and the o t h e r , x,  f i e s the c o n d i t i o n  x = 0(et)  as  0.  z  for  satis-  For the three and four-body  problems considered i n the present a n a l y s i s the equations of motion contain t e x p l i c i t l y , and i t i s therefore to set the f a s t v a r i a b l e r equal to t .  both convenient and expedient The slow v a r i a b l e i s chosen so  that x = e t , which i s the simplest case s a t i s f y i n g i n x then requires that R ( ^ , T; z) = 0(e ) n  n  that t = 0(e~^).  x = 0(et).  Uniformity  f o r x f i x e d as z -> 0, so  This c o n d i t i o n consequently determines, f o r small  values of e, the required large time i n t e r v a l during which the s o l u t i o n i s uniformly v a l i d .  The g e n e r a l i z a t i o n of t h i s procedure to the many  v a r i a b l e case and a time i n t e r v a l 0(e~ ) (where n" = 2 , 3 , ••••) i s n  described i n Nayfeh [ 4 6 ] . tf  When the asymptotic expansions f o r  u(t)  and v ( t ) are s u b s t i t u t e d i n the equations of motion secular terms appear i n the r e s u l t i n g general s o l u t i o n , and i t i s the e l i m i n a t i o n of these non-uniform terms which defines the uniformly v a l i d s o l u t i o n . With t h i s section the foundation of the development i s completed, and we can now proceed to the a n a l y s i s of three and four-body  79 3. motion.  A uniformly v a l i d s o l u t i o n of the s t e l l a r three-body problem i s  derived i n Chapter 4, which may be reduced to a corresponding s o l u t i o n of the r e s t r i c t e d problem by applying the l i m i t process m^ + 0.  In  t h i s form a comparison with G.W. H i l l ' s v a r i a t i o n o r b i t i s f e a s i b l e , and we obtain d i r e c t v e r i f i c a t i o n of the uniformly v a l i d s o l u t i o n from w e l l established r e s u l t s .  The general three-body o r b i t i s then applied to  the four-body problem, and i n Chapter 5 a uniformly v a l i d s o l u t i o n i s derived f o r motion near the e q u i l a t e r a l t r i a n g l e points of m  9  and m-.  80 4.1  4.  4.1  THE THREE-BODY PROBLEM  Introduction Equations 3.51 and 3.52,  together with Equations 3.5 and 3.6,  describe the least-complicated form of three-body motion f o r p-|, p and Pg, and the s o l u t i o n of these equations w i l l chapter.  2  be o u t l i n e d i n t h i s  The i n i t i a l a n a l y s i s presented here i s s u f f i c i e n t to reveal  the basic s o l u t i o n s t r u c t u r e , but i n the i n t e r e s t of r e a d a b i l i t y much of the d e t a i l e d development i s relegated to Appendix I I I .  It  will,  perhaps, be helpful at t h i s point to summarize .the sequence .of r e s t r i c t i o n s which generated Equations 3.51  and 3.52.  1.  m-| > m^ > m^.  2.  p.| and the hypothetical p a r t i c l e p* move i n two-body o r b i t s about the i n e r t i a l o r i g i n .  3.  p  2  and p^ move approximately i n two-body o r b i t s about t h e i r  centre of mass. From page  2.30:  4.  p* i s located at the o r i g i n of the.?;, n , C coordinate system.  5.  p-| and p* move i n c i r c u l a r o r b i t s about the i n e r t i a l o r i g i n .  6.  The motion of p.j-, p  F i n a l l y , from Section  2  3.6.1:  and p  3  i s coplanar.  81 4.2 7.  As the small parameter P  e tends to z e r o , the l o c a t i o n of  i s defined by the s o l u t i o n of r e l a t i v e e q u i l i b r i u m  2  C (t) = 1 - y 2  2  ;  n (t) = 0 . 2  Note that conditions 5, 6 and 7 combine to produce the simplest form of three-body motion. assumption.  The only approximation i s associated w i t h the second  This source of inconsistency was discussed i n Section  (Equation 2 . 3 9 ) , but as  e -> 0  The a n a l y s i s r e s t s , t h e r e f o r e ,  4.2  2.6  the r e s u l t i n g e r r o r reduces to zero. on a comparatively f i r m foundation.  Pre!iminary A n a l y s i s From Equations 3.49 and 3.50 the p o s i t i o n of  i s determined  by  where the o r b i t a l parameter 6 has been set equal to zero.  We now express  u(t) and v ( t ) as asymptotic expansions i n the two time v a r i a b l e s x and t , with  $JtO  x = e t , so that  ^  •«-/*,.•+  -T L—i  1  u^r,t)e~  4-  OC<L^)  C4 3)  82 4.3 where these expressions are to be uniformly v a l i d i n T and t . formal operation of d i f f e r e n t i a t i o n i s assumed to be v a l i d Section 3 . 7 . 1 ) , then f o r the d e r i v a t i v e s of eu(t)  If  the  (see  and e v ( t ) we o b t a i n :  C4.?>  N  6 /v/ Jib  4-  & 3y„,  5" Pb^Tr  Pb""  ^b^r  1  Equations 3.51  and 3.52 can now be expressed as a sequence of non-  homogeneous p a r t i a l d i f f e r e n t i a l equations i n ascending powers of e, For the i ' t h power of e we have <)t  2- h»  3^-  ^  c  k  £ * C u,v,  b,  v )  v  In the absence of any information about i t s v a l i d i t y , we have no a l t e r n a t i v e but to proceed i n the hope that t h i s operation i s justified.  (4.1)  83 4.4  . ¥ ( u,v b , v ) K  ;  ^t -  ,  0*--'°)  at  1  where the functions E* and F* involve ( i n general) the terms u^ to u^._^ and v-| to v^_^ but not u^ and v ^ .  Now the ordinary d i f f e r e n t i a l  equations  £  -?.</-  3u_ = o  (>•")  have the f o l l o w i n g s o l u t i o n :  where A , B,C and D are a r b i t r a r y constants.  The homogeneous s o l u t i o n s  f o r u.j ( t , x) and v ^ ( t , x) i n Equations 4.9 and 4.10 therefore assume an i d e n t i c a l form of x .  except t h a t , i n t h i s case, A , B, C and D are functions  Equations 4.13 and 4.14 can be rearranged so that  i^lt/e)  =  ZolL-v)  - izCv)c°$[t  +• 0 0 c ) ]  which, i n p r a c t i c e , i s a more convenient form.  (4.1?)  The functions a ( x ) ,  3 ( T ) , Y ( ) and 9(x) are to be determined from the c o n d i t i o n s f o r uniform t  v a l i d i t y i n t and x .  Note, i n c i d e n t a l l y , that Equation 4.16 already  4.5 contains a secular term. In the absence of f o r c i n g terms i n v o l v i n g Uy, Equations and  3.52 possess the exact s o l u t i o n  discussed i n Section 3.5.  ~ ^ ~ ^2'  I t i s therefore  n  2 ^  =  ^'  3.51  a s  reasonable to assume that any  disturbance from r e l a t i v e e q u i l i b r i u m caused by the f o r c i n g terms w i l l be of a comparable order i n e, so we can take  £  <^ ) f c  =  J " ^ ( t , T ) r Z >  +  -  OU ) n  .  (4.1?)  r>.-l  .  I,  (4. 18)  »~ =. k  where k > 1.  To determine the value of k we must now examine the non-  hnmnopnpnu?; f u n r t . i n n f ;  and  | r „ !  3  in Equations 3.51 and 3.52, which can be expanded using Equations A1.6 and it Al.7  A1.7 i n Appendix I.  As the non-homogeneous functions are O'(e^),  i s not u n r e a l i s t i c to assume that k ^ 4. we then obtain  From Equations A1.6 and  85  4.6  + 6  3. cob  3b  JS_ S  (4-.2°)  since  £  2  =  1  - y  2  + °(  e 4  )  From Equations 4.15 and 4.16  a n d  ^2  = 0  (  e  u ^ ( t , x) and  4  when  )  k = 4.  ( t , x) are p e r i o d i c i n  8  8  t = 2TT, S O that terms of the form e acost  and  e gsint  i n Equations  4.19 and 4.20 w i l l generate secular terms i n the non-homogeneous s o l u t i o n f o r Ug(t, x) and V g ( t , x ) .  I f the complete s o l u t i o n i s to be  uniformly v a l i d i n t these secular terms must be e l i m i n a t e d , which can only be.accomplished i f the q u a n t i t i e s and  8 i n Equations 3.51 and 3.52 are 0 ( e ) . This c o n d i t i o n i s s a t i s f i e d when k = 5. The same r e s u l t can be o b t a i n e d , i n a l e s s h e u r i s t i c manner,  86 4.7 using the complete expansions of Equations 4.3 and 4 . 4 , although at the expense of s u b s t a n t i a l l y increased complexity i n the a n a l y s i s .  A more  complete expansion of Equations 4.19 and 4.20 i s given in Appendix I (Equations A1.8 and A'1.9) f o r  L  j  m > 5  At the s t a r t of the sequence of non-homogeneous p a r t i a l d i f f e r e n t i a l equations corresponding to Equations 4.9 and 4.10 we have  and t h e r e f o r e , from Equations 4.15 and 4.16:  ^(t,f)  =  2yScO sc*.[ t +  9CT)]  +XCV)  -  3-bocCv)  ..  Higher order s o l u t i o n s w i l l , i n g e n e r a l , c o n s i s t of both homogeneous and non-homogeneous terms, so that we can w r i t e  Z.—x  m a t  (*-*0  87 4.8  where the subscripts H and NH denote homogeneous and non-homogeneous parts of the s o l u t i o n . solutions into  I t i s convenient to accumulate a l l the homogeneous u (t, T ) 5  h  and  v ( t , T ) , but i n t h i s case we must 5  h  express the functions a(-r), B ( T ) , y("0 and 6(x) as asymptotic expansions of the form  I.  = 2^  i  ^ ^ ^ ^  Equations 4.27 and 4.28 can then be w r i t t e n  C + -3I)  88 4.9  feu. Z . — i  yt-l  Z — - i  m-  6  Now i t i s always p o s s i b l e , by r e - l a b e l l i n g x as e t , to express the nonhomogeneous s o l u t i o n s u . and  (where i > 6) as functions of t , and  therefore no d e r i v a t i v e s of these terms with respect to x need appear i n the sequence of p a r t i a l d i f f e r e n t i a l equations.  With t h i s s i m p l i f i c a t i o n  we obtain  -a dO-5  it*  +  E  it  (4-.»0  - 5. J Oil)  -3u.  f  =  -Z  —  o(0  - 2  —  J Of.)  89 4.10 >Jt<  -  it  r-  -  his  —  oCfe) (4.31)  +  ^ +-  23u  it  J  E,  O(0  g  it 2 0 + - p,) ". 1  [5  where the subscripts 0(1),  0(e)  order i n e and the functions (Equations A1.8 and A 1 . 9 ) .  and 0(e ) denote terms of corresponding and F. are defined i n Appendix I  The extension of t h i s sequence to higher  orders of e i s given i n Appendix I I I ,  4.3  Equations A3.20 to A3.29.  The Uniformly V a l i d S o l u t i o n We can now solve the sequence of p a r t i a l d i f f e r e n t i a l  equations  under the condition that no secular or unbounded terms remain i n the solution.  A development as f a r as s o l u t i o n s f o r Ug and Vg i s  to reveal the e s s e n t i a l  sufficient  features of the a n a l y s i s , and d e t a i l s of the  f u r t h e r development can be found i n Appendix I I I . A l l non-homogeneous terms i n the sequence of p a r t i a l d i f f e r e n t i a l equations may be w r i t t e n i n the form  &*Ctp)  ~  a„  4-  P*(fc,V)  =  y  +  o  a. b x  4-  a ^ j U v b  +  b^<-b  4-  A ^ C s o t  by l o t  4-  •*•  c «i^ wfc  +  c . t o o k  <^ AV<0b  -4-  cL C o « o b  D  o  {  (4.4l)  (4.4*)  90 4.11 (the functions E* and F* are defined i n Equations 4.9 and 4.10).Corresponding non-homogeneous solutions are given i n Appendix I I I (Equations A3.10 to A3.13), and the conditions under which secular terms are eliminated w i l l determine the uniformly v a l i d s o l u t i o n . When the asymptotic expansions defined by Equations 4.29 to 4.32 are s u b s t i t u t e d i n Equations 4.25 and 4.26 we obtain  ""  6./», + fc /3*.*--- J COS £ b + 0 f £0, + e?9^ir -• ] l  o  The functions c o s ( t + 0 ) and s i n ( t + 9 ) are expanded as Cn C t + P)  -  <^b coP - <- b «u. ©„ 0  - «E> I s—t c<o (4.45)  ( t + 6)  *  sCtb c«o9 4- cr> b sJ-v. P„ + £ P, I dr»t CoPo p  —  ^1 b  P ] e  +_of O  so t h a t , f o r the s o l u t i o n of Equations 4.23 and 4.24, we may w r i t e  fr  91  4.12  (a. 4 y )  The secular term in Equation 4.48 could be eliminated by s e t t i n g CXQ(X)=0, but t h i s assumes, without any a d d i t i o n a l proof, that «Q(T) = constant. I t i s not, i n g e n e r a l , necessary to make any assumptions of t h i s nature, so we w i l l proceed as i f  CCQ(X) ^  0.  A f t e r taking the p a r t i a l d e r i v a t i v e of Equations 4.47 and 4.48 with respect to x and s u b s t i t u t i n g f o r Eg and Fg from Appendix I (Equations A1.8 and A 1 . 9 ) , we obtain f o r Equations 4.35 and 4.36:  2 <-<-<  -  ^  2s.«-b  - x  *• % SC«-b  +  4-  <ht  Z«  2v  Oil)  +  6b  2c<rz> b  5  —  2. dot  ^/3„ C«Pp  - p> ^^ 0  0  p_Pe  I f terms i n these equations are i d e n t i f i e d with q u a n t i t i e s i n Equations 4.41 and 4.42, a corresponding non-homogeneous s o l u t i o n can then be derived from Equations A3.10 to A3.13 i n Appendix I I I . example,  We have, f o r  0  92 4.13  2>V  «  '  To eliminate terms i n v o l v i n g t  }x  2  i . [ Wo*****] ;  from the non-homogeneous  = o  0  solution  (A- * >  so we are now j u s t i f i e d i n taking  *  -  0  °  to remove the secular quantity i n Equation 4.48.  C^.rz)  I f terms containing  t s i n t and t c o s t are to be e l i m i n a t e d :  P  I  ^  f f  I  =.  ©  ,  (A-.SO  1  so t h a t , i f 3 Q ^ 0 , both BQ and 6Q must be constant. 4.53 and 4.54 are s a t i s f i e d we have  When Equations 4 . 5 1 ,  93 4.14  6  u  pit '?*-}  =  ^6  1  [  "<«  X c e r > i :  -  £  ,zt  *  0  If u  6  c  b ]  0t?  Ur  i s to be free of secular terms, then  *u.t -  [  2-t - 4jiK.t + £ t  <LXo must be constant. its  Now  contains a term -e[3ta^] which, i f a-j i s constant, could be used to eliminate the secular terms i n Equation 4.56. the r e s u l t that  = o  (Equation 4 . 6 4 ) .  Suppose we a n t i c i p a t e Then, i f  the l a r g e s t secular terms i n the complete s o l u t i o n are 0(e ) when t i s f i x e d as  e -»- 0.  A l s o , i f we take  then  To j u s t i f y t h i s choice of a-j, suppose We wil1 then have  O  I  O  9k 4.15  .where TQ and  are constant, and Equation 4.57 may be w r i t t e n as  17Exactly the same c o n d i t i o n i s obtained as i n Equation 4.58, but f o r + r.j instead of a-p Y Q ( T ) "=  F-| is therefore redundant, and we are l e f t with  r , which i s equivalent to Equation 4.59. Q  The non-homogeneous s o l u t i o n s Ug(t) and Vg(t) now reduce to  ii  ?  $ul  fc. -  +  £b  4-  s o l u t i o n by the c o r r e c t choice of For Equations 4.37 and 4.38 we have, from Equations A 1 . 8 and A l . 9 i n Appendix I :  o  With the s i m p l i f i c a t i o n s r e s u l t i n g from Equations 4 . 5 1 , 4 . 5 3 , 4 . 5 4 and 4.59 we obtain f o r the d e r i v a t i v e terms:  3  u.  x  s  Otl)  4-  sc~b  V, iv  95 4.16  I at-  \  L  \  ivJ  air ar  ' 5i  a_P  c  av  L  a*r  _ «*p„ ) z^s, aj> 40  I  -  ^ v -t5  . aT^tr  a\  Ptr av  '  ar'  Z f^,  s  ar  ar  P/S  Z c « f f , J / J , £P,  *•  I  +- /5,a_P  0  I W  ?  i v )  av  J  L  at" A ^  aira r  - coj k  a"v  v <aW J  1  (.  a r a-r  '  at\) _  Secular terms i n v o l v i n g t , t s i n t and t c o s t w i l l be e l i m i n a t e d from the s o l u t i o n s f o r Uy and Vy i f :  1«,  =  o  96 4.17 (4.6$)  at?  Tv  j-c*  V a w  '  ar  a-r  s  and these conditions are a l l s a t i s f i e d when  o and 6^ are  ,  constant.  The remaining secular term i n Vg i s removed i f  atr  aV  + 3^9  av  at:  n  air  where  1  a-t" 1  I  i s the only term which i s undetermined.  airy  From the same argument  as before f o r YQ i t follows that  A l l the q u a n t i t i e s in Equations 4.62 and 4.63 are consequently e l i m i n a t e d , so that we obtain  H (fc) = V (t) ?  ?  = O  .  (4.61)  The conditions f o r uniform v a l i d i t y of the complete s o l u t i o n have, at t h i s point i n the a n a l y s i s , produced e x p l i c i t values only f o r ctg and  , and to determine BQ and 6Q we must consider the s o l u t i o n s  97 4.18 for Ug and v .  From Appendix I , Equations A1.8 and A1.9:  g  1 tosl +  ]S COS 3fc  «  8  3_s^b  +  8  If 5 ^ 3b 8  and the non-homogeneous terms i n Equations 4.39 and 4.40 can be w r i t t e n i n the form  -  J U  co^St  ix  hx  */ 60s  ]£. j * , ( l - ^ . ' )  t ix  iX  e  + jc*.b  2 C 0 O  ^ 5  t  - Zs^B a ^K 0  0  •  +  2-0+-/O A> X  ^ " ^ o  (4.?3)  98 4.19  To remove terms i n v o l v i n g t  2  from the non-homogeneous s o l u t i o n  ^ 2 . - 0  C4.?0  o and f o r the e l i m i n a t i o n of the q u a n t i t y -e [ 3 t a , J from Equation 4.44 we must take =  0.  (4-.7O  Secular terms containing t s i n t and t c o s t are eliminated i f  I  J  2V  lb  and  bx  .  16  Now, unless s i n 6Q = 0, Equation 4.77 v i o l a t e s uniform v a l i d i t y i n x , and therefore  From Equation 4.76 we have  99 4.20 and since 3  n  and 9  u  0  constant,  a r e  it  i s also constant.  I f , however,  i s also constant, then i t f o l l o w s that  2  fi  N  u  = is  0  cos 9  C*-*0  V  Although t h i s d e r i v a t i o n f o r gg and 9  cannot be j u s t i f i e d at  2  the present stage of the a n a l y s i s , i t w i l l be shown i n Appendix III  that  +  6^ v i o l a t e s uniform v a l i d i t y i n T unless 9 a p p l i e s , i n f a c t , to Qy i n Appendix I I I ) ,  2  =  0-  t n  e  same c o n d i t i o n  9 ^ , and 9 ^ (which i s the l i m i t of the a n a l y s i s  and i t appears l i k e l y that 9^ = 0 f o r a l l i > 0.  A f t e r some m a n i p u l a t i o n , the one remaining secular term i n Vg can be obtained i n the form «  when Y  2  iv  i s constant (by the argument of page  4.14), and i f  is also  constant, then the term -e [3ta.j] i n Equation 4.44 i s e l i m i n a t e d by  oi  3  =  Z  j^O-fO^  .  At present we have only assumed that 0  (4.22)  2  and  are constant.  Both these  assumptions a r e , however, v e r i f i e d i n Appendix I I I , and f o r Ug and Vg we then have  Page  A3.24,  Equation A3.83.  100 4.21  - 2_  +  4  8? c o s t 64  + «  tos  H  (4-33)  »4  31 s « t + rs s c ^ 3 b 3732.'  2-1 b  where the secular term i n v ( t ) i s eliminated by the c o r r e c t choice of g  a. This process o f a n a l y s i s , which i s continued i n Appendix 111  :  leads to the f o l l o w i n g r e s u l t s : (*•«*)  3r  ^5 *  31  -t-  &4  Z%£  26  .(4.*t)  f*,)  (4.S?) 64  0 + ^ , ) ^  S?6  II.  64  (4.88)  (4.89)  141  +  135 3z(  p-, l +  (4.1.  .p >J l  (4.12.)  O  u^(t) = f*,('+^,) (i-j*0 i  )  -9- +• M t o j t £>  -  i_5 c o s 2-t 6  (4-«3>  101 4.22  ?>  17.  (4.14)  2.  2 +• 22-5" p. i  1 v ft) 10  ^,(i- / O | -T  =  4  +  -63  -  i  7  Go2t  ^ /"*• 7 '*"  ~ •(  Jt  si^i+b  3 ^ t - s ^ . k )  -  t  •f 15' I 64-  +  6  +  7  +  t^i  x % s  ? t y.xCj+w,) >  7  A-t  Co5  S  ^  Six. 2.b  1j  <eS/*,(l-jO ^ ( l - c a t " }  (A-.9fc)  16 C.\.+.i*, ) >  (. - f O  where the development i s terminated at O ( e ^ ) .  Note that the  secular  terms i n Equations 4.94 and 4.96 are eliminated i n the complete s o l u t i o n f o r ev(t) when  and a,- s a t i s f y Equations 4.85 and 4.86 r e s p e c t i v e l y . 3Y  No s p e c i f i c conditions apply to y » except that ^± = 0, so ax that y appears to be an a r b i t r a r y constant q u a n t i t y . no reason to exclude the c o n d i t i o n are v a l i d f o r 0 < ^  < 1.  ->- 1 , since the equations of motion  In t h i s case Equations 4 . 8 6 , 4 . 8 8 , 4.95 and  4.96 contain unbounded terms unless Y  q  and Y-J are both z e r o , and i t  appears l i k e l y t h a t , i n g e n e r a l , Y ( T ) = 0. When Y s t a t e from Equation A 3 . 2 4 1  There i s , however,  that  = Q  0 we can also  102 4.23  The values given i n Equations 4.85 to 4.97 are e n t i r e l y l e g i t i m a t e , i n that no assumptions whatever are necessary.  If,  however, we make the  p l a u s i b l e assumption that ctg, 6g and 6^ are constant,  then we may also  wri te  (from Equation A3.250 with YQ  =  Y-j  =  0)  ;  5*c  Sec  e,  0„  ( A . too  In a d d i t i o n to these concrete r e s u l t s , the d e r i v a t i v e s of a , 3, y with respect to T are at l e a s t O(e^), ||- = O(e^) and  tP  =  0(e°)  From Equations 4.85 to 4.100 constant),  solutions for ^ ( t )  a constant e r r o r O(e^).  and r ^ t )  can be accumulated that have  Secular terms i n the s o l u t i o n s f o r 2  and v ^ ( t )  ( and assuming 8g and 0^ are  could involve t , t  u^(t)  3 and t  expressions  (see Appendix I I I ,  Equations A3.8 and A 3 . 9 ) , but i f secular terms are eliminated from the  )  103 4.24 lower-order s o l u t i o n s no t  J  expressions w i l l be generated.  I t a l s o appears  2 highly l i k e l y that t  terms are e l i m i n a t e d i f a ( r ) = constant.  With a  reasonable degree of confidence we can therefore state that the s o l u t i o n s for ^ ( t )  a n c  '  a r e  u n i  ^  o r n i  ^ y v a l i d f o r t = 0 ( e ~ ^ ) , but even the  most p e s s i m i s t i c estimate gives a time i n t e r v a l of uniform v a l i d i t y 0(e" ). 7  A f t e r some m a n i p u l a t i o n , these s o l u t i o n s can be derived as:  J. + Cos  -  cos  -  ^  It  3 fc  &4-  (4. e ^ / l f ^ ' i M i - ^ )  - £  ID  l o S d - ^ )  3  c^t  + 1 1  'oO  cos 2.1 ]  co^ 4-t  +  IS ^ Ci-p,_") sc*.b (  *  +  .b  £ t  l f o  •  Ifed-t-fJ,) J  -  tS"  sin.  32-  J  3b (4.IOZ ) -  »2-  101}. 4.25 Zb  +•  fe3  ScV4b  10x4 Ci+^i 7) 3 + ocv)  ,  where i t i s assumed, i n Equation 4.102, that y{x) = 0(e ).  Note that  any ambiguity concerning the choice of 8Q i s resolved i n t h i s f i n a l form of the s o l u t i o n (see Equation 4 . 7 8 ) . . Corresponding r e s u l t s f o r £ ( t ) and n ( t ) 3  from Equations 4.101  3  can r e a d i l y be obtained  and 4.102 using Equations 3.5 and 3.6.  We then  have:  > A<ifc.£'-/0 s  /6 (1+  -r  6  U  ' z? + 3 2  -  >35  pi,  est  do  6  2b  ) ~* - 2£  ? «jt  3^1+^1) J  io5(l-p^")  +  cost  cos 3 fc  ^4-  +  co^2.b  n  (4.  *  103)  cos4b  32.  OCfe")  ^ 0  11  Si*>.t  — _i£  SCn. 3 b  32.  -+L  640+/X,")  SH  12.  Si«,?-b  J  ]  sc^zb  (4. 1 0 4 )  105 4.26  4.4  A Uniformly V a l i d S o l u t i o n of the R e s t r i c t e d Problem Quite a r b i t r a r i l y we have chosen m^ > m^, so that  Now suppose U £ •+ 1 o r , conversely,  0.  >  In t h i s case Equations  4.103 and 4.104 reduce to a s o l u t i o n of the r e s t r i c t e d problem, but i n a coordinate system where the p o s i t i o n of m-j changes with time.  When  U £ -> 1, Equations 4.103 and 4.104 take the f o l l o w i n g form: r L  6  O f t ' )  (4.«oS)  1  /  \!  i±  Si*, i t  12.  9  (A-.lofc")  where these s o l u t i o n s are uniformly v a l i d as e -> 0 f o r t = 0(e" Note that  Ur*.  $ (t)  = o  t  ^ Ct)  - o  ,  ). and  consequently m i s located at the o r i g i n of the £, n. coordinate system. ?  106 4.27  Figure 4-1  Primary c o n f i g u r a t i o n f o r the r e s t r i c t e d problem  I f we s e l e c t the H , H coordinate system shown i n Figure 4 - 1 , so that both  and m are located on the E a x i s , then the transformation 2  equations are (4.|o?)  H  =  ^ d.ib -<•  cos t  (A--log)  With respect to the ~, H axes, Equations 4.105 and 4.106 can now be w r i t t e n as:  C fc) = .  cos b  4-g  it  *4-  2  107 4.28  -  - Sin.t 4?  A  3-4  (4.l«o)  6  8  if.  In t h i s form the s o l u t i o n f o r the motion of  can be compared  with H i l l ' s v a r i a t i o n o r b i t , which we consider i n the f o l l o w i n g s e c t i o n .  4.4.1  H i l l ' s s i m p l i f i c a t i o n of the r e s t r i c t e d problem G.W. H i l l based his a n a l y s i s of lunar motion on a three-body  model which corresponds to a s p e c i f i c s i m p l i f i c a t i o n of the r e s t r i c t e d problem with the sun and earth as primaries  [52 * ] .  A  discussion of H i l l ' s a n a l y s i s can be found i n Brown [ 5 3 ] Hagihara [ 5 4 ] ;  detailed  and  f o r a concise summary see Brouwer and Clemence [ 4 2 ] . 3  The p r i n c i p a l assumptions of H i l l ' s approach a r e : the s o l a r p a r a l l a x i s zero .t  1.  More s p e c i f i c a l l y the a n a l y s i s requires that a / r ' be n e g l i g i b l e , where a and r ' denote the mean earth - s a t e l l i t e and earth - sun distances r e s p e c t i v e l y . I f the e a r t h ' s equatorial radius i s denoted by r * , then the lunar and s o l a r p a r a l l a x TT and T T are defined by: o  = and, since r a_ r'  L  arc s i n «  a «  a  arc s i n  _r r'  r'  7L-  C I f the s o l a r p a r a l l a x i s z e r o , a / r '  i s therefore zero.  108 4.29 2.  the s o l a r e c c e n t r i c i t y  i s -zero;  3.  the lunar i n c l i n a t i o n i s z e r o ;  4.  m /  i s s u f f i c i e n t l y small "to be neglected.  x  •  / lA |  Note that although H i l l ' s a n a l y s i s does not e x p l i c i t l y require m^ to be n e g l i g i b l y s m a l l , t h i s condition i s implied by s e t t i n g the e c c e n t r i c i t y equal to zero.  solar  The sun's o r b i t about the e a r t h ' s mass  centre i s consequently c i r c u l a r , so that m-j and f i g u r a t i o n independent of the motion of m^.  maintain a f i x e d con-  Note also that the Jacobi  i n t e g r a l , which H i l l uses to e l i m i n a t e nonlinear terms from the equation of motion, i s i n v a l i d when m^ d i f f e r s s i g n i f i c a n t l y from z e r o , as indicated i n Section  2.7.  Kevorkian [55 ] has demonstrated, using an elegant and d i r e c t nieuiGu,  Midi, uic r t i u i c u e u |j I  UU  t ein can ue  ICUULCU  l i m i t process to give H i l l ' s equations of motion. also appears i n Szebehely [ l  1 0  ],  uy an as,ym[j lui, i C A s i m i l a r approach  but i n the summary given here we f o l l o w  Kevorkian i n l o c a t i n g the l a r g e r mass (m-j) on the negative H a x i s (Figure  4-2). The c o n f i g u r a t i o n of the planar r e s t r i c t e d problem i s shown i n  Figure 4-2, where the larger mass i s associated with the sun and JL  -  ^^/^  •  The time and distance normalizations are s e l e c t e d  so that the H * H* coordinate system rotates at u n i t angular v e l o c i t y w i t h respect to i n e r t i a l axes, and the earth-sun distance  is unity.  We next locate m^ at the o r i g i n of a new system by s e t t i n g *  =  <j  =  -  H  ' 0 - / 0  .  . .  (4.ni>  (.4.WZ.)  109 4.30 ri  Figure 4-2  Primary c o n f i g u r a t i o n , with respect to the H * , H* coordinate system, f o r the r e s t r i c t e d problem.  i n which case m^ i s located at x = - 1 .  The equations of s a t e l l i t e motion  can then be w r i t t e n as:  &  - 2..  at +  +  l -  r.3  4-  i - 0-/O  where  An asymptotic expansion of the form  p.  X  110 4.31 C4-.ll?)  (4.  i s not v a l i d near the smaller primary, as the g r a v i t a t i o n a l J i  and  us)  terms  appear as 0(y) whereas in p r a c t i c e ,  for  a r b i t r a r y u, there e x i s t s a region i n the neighbourhood of m w i t h i n which 2  these terms dominate the equations of motion. defined by Equations 4.117  and 4.118  Near the o r i g i n a s o l u t i o n  i s consequently  not uniformly v a l i d  i n the space v a r i a b l e s x and y as u -»• 0. This non-uniformity can be avoided i f we take t  and, i n terms of these new v a r i a b l e s ,  +  «  t  U.II<O  Equations 4.113  and 4.114  can be  wri tten o(.-3-/3  i  -  2.  ^  s (4.U0)  i - Ci->) dLk  a  dt  where, f o r convenience of n o t a t i o n , t q u a n t i t i e s r^ and  now take the form  r  3  - ^ r  (4.13.1) 3  has been replaced by t .  The  111 4.32  so t h a t , provided a > 0:  y.->  o  At t h i s point i t i s only necessary to consider one component, and i f we s u b s t i t u t e  JLt  for  i n Equation 4.120  using Equation 4.123:  1  (4.12 4-) '4  The values of a and 3 can now be selected from the c o n d i t i o n that s p e c i f i c terms i n Equation 4.124 If,  are of a p a r t i c u l a r order i n y .  f o r example, s a t e l l i t e motion i s predominantly determined by  planetary g r a v i t a t i o n ,  then:  I - 2-OL  =  O  112 4.33 so that a = 1/2,  6 = 1/4.  This case, where s a t e l l i t e motion takes place  in the . v i c i n i t y of m^, i s considered i n d e t a i l by Kevorkian [55 ] .  Con-  v e r s e l y , when the planetary g r a v i t a t i o n , C o r i o l i s and c e n t r i f u g a l accelerations  are a l l of the same order we have:  oi - %fi  and  =  dL-p  therefore a = 1/3,  4.120 and 4.121  d-t  I f terms 0(\i%)  3=0.  With these values f o r a and 3, Equations  can be w r i t t e n as:  r  ,  ,-,3/,  are neglected and we set  (see Equation 4.123 with a = 1/3),  Equations 4.125 and 4.126  i d e n t i c a l to H i l l ' s equations of motion [ 5 2 ] . 2  are  In the terminology of  asymptotic expansion theory, H i l l ' s equations are therefore obtained from Equations 4.113 and 4.114 the v a r i a b l e s  f o r the planar r e s t r i c t e d problem by holding  113 4.34  f i x e d as u -> 0. The p a r t i c u l a r symmetric p e r i o d i c s o l u t i o n of Equations 4.125 and 4.126  corresponding to the i n i t i a l  5 / 0 )  >  o  ;  ^Co)  -  defines H i l l ' s v a r i a t i o n o r b i t [ 5 2 ] . 3  conditions  o  j  5^°^  =  0  This s o l u t i o n i s compared with  the o r b i t defined by Equations 4.109 and 4.110 i n Section 4.4.2 below. »  4.4.2  H i l l ' s variation orbit From the general s e r i e s s o l u t i o n , H i l l computed a p e r i o d i c lunar  o r b i t assuming the f o l l o w i n g sidereal periods  T  L  =  27.321 661  T  $  =  365.256 371 .  [l  1 1  ]:  Here T^ and T<. denote the lunar and s o l a r periods r e s p e c t i v e l y , and these numerical values agree with the observed periods i n ephemeris days f o r the earth-moon and earth-sun  systems.  +  S l i g h t l y d i f f e r e n t values are i n current use: Sidereal mean motion of moon (1900) i n r a d i a n s / s e c . E.T. [ 5 6 ] 2.6616 99489 x 10"  K  Sidereal mean period of moon (1900) .in ephemeris days  27.321 66140  Sidereal year 1900 i n ephemeris days [45 ]  365.256 36556  D  114 4.35 The corresponding v a r i a t i o n o r b i t , which appears i n Brouwer and Clemence [42 *] i n a form compatible with Equations 4.109 and 4.110, 1  i s given below. X  Y  =  =-  o. <Ho4o  s?-&og  cost  • 4 o. ooooo  SS?$8  COS. St  •*• o. ooooo  ooioo  cos ? t  + o. ooooo  00007.  i, C O 7 7 8  iso-il  •» o. oo 151  4i  + o.  oo o o o  56 7  •*• o .  4 0*  CDS 'J  C*4. I Z 7 )  fc  sc^t  13  sc«. 3b  09  S"fc  ooooo  06300  s^7b  ooooo  00007.  s;«. 1 1  (4.iae)  I t i s assumed, f o r t h i s s o l u t i o n , that new moon occurs at t = 2nTr (|n| = 0, 1, 2, ••••) and that the l a r g e r mass m^ i s located on the positive X axis.  Equations 4.109 and 4.110 were derived f o r the same  i n i t i a l c o n f i g u r a t i o n but with m^ on the negative H a x i s (see 4-1). H = -Y  Figure  The two o r b i t s should therefore be equivalent i f we set E = - X , so that Equations 4.109 and 4.110 take the form:  115 4.36  X(b)  =  cost  1  -  65 46  + cos 3fc  3  .+ 7 6*p,(i+n,Y  16  «  *4  4-8 9  6  1  Now from Eauation 3.21  where fi = T^/T^.  Consequently, f o r the s i d e r e a l periods used by  Hill:  9. =  7.4801 32633 x 10  -2  The earth-sun distance can be obtained by expressing the a . u . i n dimension-free form. 1 a.u.  =  From A s h , Shapiro and Smith [57 ] : 4.9900 4785 x 1 0  and from Clemence [56 ] :  2  L i g h t seconds,  116 4.37 Mean earth-moon distance i n metres  q A = 0.384400 x 10  V e l o c i t y of l i g h t i n metres/second  q c = 0.2997925 x.10  .  We then have the f o l l o w i n g numerical values f o r r-j and u-|: r  1  =  3.8917 24558 x 10  2  y  ]  =  3.2979 46189 x 10  5  ,  and from Equation 3.28 f o r e:  e  =  5.0690 77750 x 1 0  - 2  .  Equations 4.129 and 4.130 can now be evaluated, and we obtain the r e s u l t s given below.  1  -  16  7.5768  6*38*1  -  3  l.6llS  5862?  % \0~*  (4.13O  S  L-.P-VW  1  ^ l o "  +  5". 7 1 1 7  *..lo"  8ZS6?  y. l o "  3  •+•  3  +  3.66ZI  l. M i S  373&S  S"€6S8  *  l o  * 10"*  -  3  (V(33)  117 4.38 xct)  =  o. <?<) O ? V  i^gl  co t 5  (4.  + OCV')  •+.O.. o o . l A - l  S3i 7e  _s;~-.3t  v  + 0 ( t " )  (4.(35)  A comparison of H i l l ' s s o l u t i o n (Equations 4.127 and 4.128) with Equations 4.134 and 4.135  shows reasonable agreement.  vergence of i n d i v i d u a l c o e f f i c i e n t s 3 Vz quantity e u-j ;  i n the present case  Note that the con-  i s determined approximately by the e  3 'A  = 0.07480, and consequently  higher-order terms w i l l have a r e l a t i v e l y s i g n i f i c a n t e f f e c t .  4.4.3  Further terms in the s o l u t i o n I t i s evident from the general behaviour of Equations  and 4.106 that the next c o n t r i b u t i o n s w i l l be 0(e  4.105  ), and although a  12 complete determination of 0(e  ) terms i s not f e a s i b l e ,  u-j (t) and v - j ( t ) can be evaluated i f we assume 2  2  case Equations A3.206 and A3.207 can be w r i t t e n :  the functions = o  •  In t h i s  118 4.39 766  ^ f l - j O  4£ 16  (Vl3<0  4  67J- p ^ O - j O cos 4fc  ccs 3 b  175  17-8  -  s.>-2.b  32.  6  4.  115 / * | C " ' - f O  32-  12.9  where we have taken YQ = Y-J  =  scw$b  12-8  Y 2. 0 . =  From Equations A 3 . 8 to A 3 . 1 1 , the corresponding s o l u t i o n s f o r u^(t)  and v ^ ( t )  are:  23-6? 768  +  (OS «>5b 48  - 13-7- -t-. ?£_1  COJS t  4e<r?2-b  + j-5  c-n? 4 b  2-^6  -  137  (U. OS)  cosZb  I%  -4/ ...  3  x  +  MS  cos b  +  JZ4S c c j 3 t i<*-4  This s i m p l i f i c a t i o n i s discussed on page  4.22.  <  +  44!  <^s S " t  119  4.40  5  «3L? t  -  5is s^b  4-  .1  £>4  ^4-  2-^6  4-S  -t- ^  sivv^-t  ? x  3072.  %S(>o  ( 4 . 13*1)  where the secular terms i n Equation 4.139 are eliminated when Equation  satisfies  4.88. The c o n t r i b u t i o n s to Equations 4.103 and 4.104 f o r £ ( t ) and 3  n-jCt) therefore take the form:  -  \$5\  e. L  +  105 4S  ?-3°4  ZfJ  cost  coot  -  1  M5 11-8  -t  4-cco2-t  +  cos4t  ;z$&  137 &o 2. b 18  Ceot  +  3-4 S ie>2-^-  cos 3 t  •+  441  cos *t  3t>7l.  (4.1*0)  120 4.41 '1  fx  6  |_o w b  r>  -t-  i  3 £ s^Z-t  - _ 2 £ s^ij-b  4-g  1  S j 8 5in.b 1  + e  .2-5  7?-  sc-vb  +  77o  sc.v3b  -t-  3'*) °i  (4.I4l)  Sivv-5t  Note, however, that these expressions are incomplete because the terms fxf>^  Co  and  $t  from  and  - 1 fxf>^.  n a v e  D e e n  omitted  r e s p e c t i v e l y (see Equations 4.25 and 4 . 2 6 ) .  The e f f e c t of t h i s omission i s purely a matter f o r s p e c u l a t i o n , but from previous s o l u t i o n behaviour i t seems probable that constant and cost terms should be removed from A , and s i n t terms eliminated r  from A  t  I f , t h e r e f o r e , these terms are neglected, then applying  the l i m i t process (1 - l ^ )  -.1  V b )  .+  0  to  and  we o b t a i n :  CA.14-2.) b  Co5 2 b  ^5 2S6  e'  7 -  ft, " 1  cos  4h.  Compare Equations 4.93 and 4.94 with 0(e ) terms i n Equations 4.101 and 4.102. Note also that the remaining terms i n cost and s i n t involve the f u n c t i o n (1 -^V» and as ( l - j ^ ) these w i l l a l s o be e l i m i n a t e d . 0  121 4.42  Cfc)  = - s>v^t  (4.143. ) 2.56  Equations 4.109 and 4.110, d e f i n i n g motion with respect to the E , H axes, can now be revised as shown below. z:  ct) =  -cash 48  *4-  cc-s 3 1  -  SIS  L  £ "-p, " cos 5"t  S  [_ 144-  4-S  I t  J  o a r )  1  J  (4.  2-St  H3Ct) = -  S<«.t  l*S  /6  £ 5  2-^  G,l^y  <S  Jin.  I  Ur  St  (. 4-S  144  •  ?t  3  <?4  (4.14?)  122 4.43 I f we evaluate the a d d i t i o n a l , c o e f f i c i e n t s values f o r e and y-j given on page  |  114-77  L  _  I-*? p,  L  42  ^£  Z  <?6  ~l  =  2.4/708  3 2 - 0 IS  *.  .3.0572  7?fc?7  * lO  (.4.146)  \0~**  ~>  e ^^," 1  4 . 3 7 , then  3  16  144  using the numerical  =  1  (  _ f c  . . ...(**. I 4 « ) t  i n which case Equations 4.134 and 4.135 f o r X ( t ) and Y ( t ) can be revised as:  X(fc)  = t  O . <no4<3  . 441S7  o . O o ! 54  ^ 3 4-1P  cos3t  3e>S72  tc. St  + o.  ooooo  cost  s  C4I49)  +oa' ) 5  Yet)  =•  I . oo765 -+ o.  001S4  + 0 . ooooo  04305  si-t  S34TO  s^3t  3e>5 72.  5i«-?t..  123 4. This s o l u t i o n shows a c l o s e r correspondence with H i l l ' s v a r i a t i o n o r b i t that i t s predecessor  (Equations 4.134 and 4.135).  must, however, be emphasized that Equations 4.149 and 4.150  It  are based on  an incomplete a n a l y s i s and that the conjecture applied i n d e r i v i n g Equations 4.142 and 4.143 cannot be s u b s t a n t i a t e d .  The r e s u l t s of  t h i s section should, i n consequence,  informed  be regarded as  estimates.  4.5  Discussion In contrast to H i l l ' s approach, which a p p l i e s only to the  r e s t r i c t e d problem, the a n a l y s i s of three-body motion presented here i s v a l i d f o r any system s a t i s f y i n g the conditions o u t l i n e d i n Section At l e a s t i n p r i n c i p l e , using t h i s method we should be able to compute lunar o r b i t s of high accuracy, p a r t i c u l a r l y when the mass r a t i o m^/m^ i s s i g n i f i c a n t l y greater than zero.  As mentioned on page  4.29  H i l l ' s s o l u t i o n i s no longer v a l i d i n t h i s case because the Jacobi i n t e g r a l does not e x i s t .  For the earth-moon system m^/ny i s  substan-  t i a l l y greater than z e r o , so that some inaccuracy might well be a n t i c i p a t e d when Equations 4.127 and 4.128 are used to describe lunar motion.  We can, i n f a c t , estimate the e r r o r introduced, by comparing  Equations 4.109 and 4.110 f o r the r e s t r i c t e d problem w i t h a c o r r e s ponding s o l u t i o n derived from Equations 4.101  to 4.104, as shown  below. The p o s i t i o n "of nu r e l a t i v e to m i s determined by: 9  4.1  121+ 4.45 - 1.  ,- e'  tost  +  e p, 6  .J. .+ c&si-b 6  Sirvb * cn-p,y*. J (4.  I f 5 , H axes are chosen so that the o r i g i n i s located at on the negative E axis (as i n Figure 4 - 1 ) ,  IS *) -  and m-j l i e s  then from Equations 4.107  and 4.108 we o b t a i n :  .3. CO  =  TTT7"  -  L  7T-  c^xt  £5  it  3. e - n,.**s 3 t + 6  o( *) £  J  1..32CI.4- p,)s-  L  (4-1*3)  J  ^  '*;<,(•-/O L  $2- C H - H . ) i J  ( 4 . 154)  1 25 4.46 where these equations are compatible with Equations 4.109 and 4.110. The earth-moon mass r a t i o deduced from radar observation i s given by Ash, Shapiro and Smith [57 ] as:  m /m 2  =  3  81.3024  +  and t h e r e f o r e :  (1 - u )  =  2  1.2150 31396 x 1 0 "  2  .  We can now evaluate those terms which do not appear i n Equations 4.109 and 4.110, and using the previous values of e and y-j (from page 4 . 3 7 ) :  e?  =  3.Z840  ^6S"<?6  x  io~  6  ,  (4-.i56)  Neglecting the mass of m^ w i l l therefore introduce errors of approximately 4.3788 x 1 0 " and 1.0947 x 1 0 " 6  components r e s p e c t i v e l y .  6  i n t o the 5 and H  S i m i l a r errors can be a n t i c i p a t e d i n  Equations 4.127 and 4.128, which define H i l l ' s lunar v a r i a t i o n o r b i t ,  +  A s h , Shapiro and Smith derived values f o r a number of astronomical cons t a n t s , using both general r e l a t i v i t y and Newtonian theory to determine a model of planetary motion. The value of the mass r a t i o quoted here corresponds to Newtonian theory.  •1  26  . 4.47  although i t should be noted t h a t a comparison of H i l l ' s o r b i t w i t h r e s u l t s derived from the present uniformly v a l i d s o l u t i o n may be misleading.  H i l l ' s i n i t i a l conjecture concerning symmetry of the v a r i a t i o n  o r b i t [52 ]  i s , f o r example, neither necessary i n t h i s a n a l y s i s n o r ,  tt  i n f a c t , evident i n Equation 4.153.  A d e t a i l e d r e v i s i o n of H i l l ' s  approach can be found i n Eckert and Eckert [58 ] , where the v a r i a t i o n o r b i t i s determined assuming: 1.  the mean s o l a r distance  2.  the s o l a r e c c e n t r i c i t y  3.  the lunar i n c l i n a t i o n i s zero;  4.  the mass r a t i o  + M  (  +  M . ^  r-| -* °°; i s zero;  + 0.  " 3 +  j  rvv  Their a n a l y s i s consequently takes i n t o consideration the combined mass of the earth and moon.  J a c o b i ' s i n t e g r a l i s s t i l l , however, a p p l i e d to  reduce the equations of motion, so that i n e f f e c t the lunar mass must be n e g l i g i b l e f o r the r e s u l t s to be v a l i d . The a n a l y s i s of Section 4.4 i n d i c a t e s that the convergence Equations 4.101 e^u-| , where  of  to 4.104 depends, to a s i g n i f i c a n t e x t e n t , on the q u a n t i t y j-i,  and  £  =  y •/  We should therefore expect p a r t i c u l a r l y convergent s o l u t i o n s f o r r e l a t i v e l y small values of u-j i n combination with large values of r-j.  This s i t u a t i o n  would a p p l y , f o r example, when two s t a r s forming a c l o s e binary o r b i t a comparatively d i s t a n t s t a r , the masses being of s i m i l a r magnitude. from page  2.26,  that provided  Note,  TX- « 1 we are j u s t i f i e d i n assuming ^ l two body o r b i t s f o r m-| and the mass-centre of m and m This s p e c i f i c r  9  v  127 4,48 case w i l l be given more d e t a i l e d consideration i n Chapter 6.  Next, how-  ever, we apply these general three-body r e s u l t s to the four-body problem.  128 5.1  5.  5.1  THE FOUR-BODY PROBLEM  Introduction Now that a uniformly v a l i d s o l u t i o n f o r the motion of p-j, p  and p  3  2  has been d e r i v e d , we can proceed to i n v e s t i g a t e the motion of p^  as o u t l i n e d i n Section 3 . 6 . 2 .  This three-body s o l u t i o n i s based upon  the c i r c u l a r ( r e l a t i v e e q u i l i b r i u m ) s o l u t i o n of the two body problem. S i m i l a r l y , the a n a l y s i s of four-body motion takes as i t s foundation e q u i l i b r i u m s o l u t i o n s of the r e s t r i c t e d problem. the r e s t r i c t e d problem can be found i n Szebehely's [1  A d e t a i l e d treatment  of  authoritative text  ] , and only those aspects which are necessary f o r the  subsequent  development are described here. Although the method of s o l u t i o n i s b a s i c a l l y i d e n t i c a l to that adopted f o r the three-body case, the a n a l y s i s of four-body motion i s s i g n i f i c a n t l y more complicated.  If, f o r example, the uniformly v a l i d  s o l u t i o n f o r a four-body o r b i t i s to have an e r r o r ©(e ), then i n 1  p r i n c i p l e the motion of p-j, p 0(e  ).  2  and p^ should be known to an accuracy  In the present case t h i s four-body s o l u t i o n i s  consequently  o  l i m i t e d to an e r r o r 0(e ) by the r e s u l t s of Chapter 4.  The a n a l y s i s  should therefore be regarded, not as exhaustive, but as p r o v i d i n g the basis f o r f u r t h e r development. +  This appears to be a feature of the expansion process not found i n the one degree of freedom case [51 ] . Note that i n the previous chapter i t was necessary to continue the a n a l y s i s as f a r as e r r o r terms 0(e '*) to define a uniformly v a l i d s o l u t i o n w i t h i n an e r r o r 0 ( e " ). A s i m i l a r l i m i t a t i o n a f f e c t s the present d e r i v a t i o n .  •129 5.2 5.2  The R e s t r i c t e d Three-bo.dy-Problem In Section 3.3 we considered the process by which the f o u r -  body problem could be reduced to a r e s t r i c t e d problem i n v o l v i n g p^, Pg and p^.  The motion of p^ i n the r e s t r i c t e d problem i s described by the  f o l l o w i n g equations:  (Equations 3.25 and 3.26).  Equations 5.1 and 5.2 possess  f i v e homo-  graphic s o l u t i o n s of r e l a t i v e e q u i l i b r i u m , and i t i s therefore  feasible  to i n v e s t i g a t e motion i n the v i c i n i t y of these points of e q u i l i b r i u m . I f we set  then Equations 5.1 and 5.2 can be w r i t t e n as  1  Mr.  -  ~ hit:  -  o  130 5.3 1  since  = n.^ = 0  = o  C5.G)  (from Equations 3.38 and 3 . 3 9 ) .  To cast Equations 5.5 and 5.6 into a t r a c t a b l e some method of expanding the nonlinear functions in x and y .  form we require  _J  as s e r i e s  I t i s , however, important that the f i r s t few terms of these  expansions should represent the nonlinear functions accurately w i t h i n a region containing the complete s o l u t i o n .  For the case of perturbed  two-body motion considered i n Section 3.6.1  t h i s region of motion i s  s m a l l , and binomial expansions are adequate.  S i m i l a r l y , f o r motion near  Lagrange points of the r e s t r i c t e d problem, a binomial expansion of the »-» r\ e> I ~i  -> v»  the numerical work of Kolenkiewicz and Carpenter [ 3 0 ]  indicates  motion i s not, i n g e n e r a l , confined to such a small r e g i o n .  that  We consequently  require an expansion which p r o v i d e s , with r e l a t i v e l y few terms, an accurate representation  inside a s i g n i f i c a n t l y larger region.  The nonlinear functions can be rearranged I  as:  +  4-  where Now  so that  l -jl r  L  denotes the distance between p^ and a Lagrange p o i n t .  131 5.4  Cs.l) Equation 5.7 can therefore be w r i t t e n i n the f o l l o w i n g form:  (5.|o)  I f we set  _L  r  i  +  ^c^ ) b  *> then a general s e r i e s expansion of the f u n c t i o n [1 + p ( x , y ) ] interval -a 6 p ^ g  •3/4  on an  can be w r i t t e n as  Cs". i i )  In p r a c t i c e the summation i s terminated at some f i n i t e value of i , l e a v i n g a r e s i d u a l e r r o r which i s , h o p e f u l l y , s m a l l .  The constant  coefficients  k . w i l l then, i n general, depend both on the type of expansion and the terminating value of i .  We s h a l l return to t h i s subject l a t e r , but at  present the general form given i n Equation 5.12  is  sufficient.  The foregoing d i s c u s s i o n a p p l i e s to motion which takes place i n s i d e a region of f i n i t e s i z e .  As i n d i c a t e d on page 5 . 1 ,  however,  s o l u t i o n s of the l i n e a r i z e d r e s t r i c t e d problem are c e n t r a l to the a n a l y s i s of four-body motion.  Consider, t h e r e f o r e ,  the e f f e c t of reducing t h i s  region to an i n f i n i t e s i m a l s i z e , so that nonlinear terms i n v o l v i n g 2 x , xy, y  2 and higher powers of x and y can be neglected.  We then have,  132 5.5 f o r Equation  i^  5.10:  I,  Equation 5.12 can be s i m p l i f i e d as:  and consequently,  from Equation  5.13:  A l i n e a r i z e d version of Equations 5.5 and 5.6 can now be d e r i v e d , using the expression given i n Equation 5.15 to s i m p l i f y the functions.  A f t e r some rearrangement  nonlinear  the l i n e a r i z e d equations of motion  can be w r i t t e n  —  X-  =  o  133 5.6  where  and Hg have been retained to preserve symmetry.  Now the Lagrange  ^ -  points s a t i s f y  — o >-3  =  O  0.1%)  (see Section 3 . 5 , page 3.17 ), and i f we set  (5-22)  134 5.7 Equations 5.16 and 5.17  take the f o l l o w i n g form:  =  CK-O  r I  ^ 3  •  t  3  1  Motion w i t h i n an i n f i n i t e s i m a l l y small region about a Lagrange e q u i l i b r i u m point i s now defined by the s o l u t i o n of these  equations.  Note that kg = 1 f o r a binomial expansion, which then reduces Equations 5.23 and 5.24  to the conventional  v a r i a t i o n a l form  [l  1 3  ]  (/.  The homogeneous s o l u t i o n of Equations 5.23 and 5.24  at)  i s derived  i n Appendix IV (Section A 4 . 1 , pages A4.1 to A4.7), but the character of t h i s s o l u t i o n depends on the p a r t i c u l a r Lagrange point to which i t r e l a t e s . There a r e , i n f a c t , four d i s t i n c t classes of s o l u t i o n , each of which must be treated (i)  separately. At the col l i n e a r Lagrange points the homogeneous s o l u t i o n of the v a r i a t i o n a l equations can be expressed as  :<-b)  A; e  B: e  Cs.zf)  Mb  C5.28)  135 5.8  where, i n g e n e r a l , there w i l l always be one unbounded term [1 (ii)  1 k  ].  Close to the e q u i l a t e r a l  t r i a n g l e points the  homogeneous  s o l u t i o n i s bounded when  r  <  2-  -J-  - 4-(k +k,)  3k*  (see Appendix IV, Section (iii)  0  A4.1)  When  V-3  _\_  —  -X - . 4-Ckp + 3 k.  O  C5-.30)  1  the v a r i a t i o n a l s o l u t i o n about  and  contains  terms of the form t s i n t , t c o s t which, i n g e n e r a l , the s o l u t i o n unbounded [ l (iv)  1 5  secular render  ].  If  hi  (S-31)  17-  3k  the s o l u t i o n near the e q u i l a t e r a l  t r i a n g l e points  and Lj.  contains terms of the form e ^ cos(cot + 8 ) so t h a t , i n g e n e r a l , ±  t  the s o l u t i o n i s again unbounded [ I  1 6  ].  The appearance of unbounded terms i n the l i n e a r i z e d s o l u t i o n s i n d i c a t e s that motion cannot,  i n these s p e c i f i c cases, be confined to  an i n f i n i t e s i m a l region about the points of e q u i l i b r i u m . may however, s t i l l  Bounded motion  be possible i n c e r t a i n instances when nonlinear  136 5.9 e f f e c t s are taken into consideration  [I  1 7  ].  In the f o l l o w i n g a n a l y s i s of four-body o r b i t s we consider only those values of y  3  f o r which Equation 5.29 i s s a t i s f i e d .  As e -»• 0 the  equations d e s c r i b i n g four-body motion reduce to corresponding equations f o r the r e s t r i c t e d problem near to use  and L ^ .  We s h a l l consequently continue  and Lg as points of reference, although i t should be noted  that i n the four-body case no p o s i t i o n s of e q u i l i b r i u m a c t u a l l y e x i s t . 5.3  Four-body Motion Near  t  and  For the p a r t i c l e p^ l e t  where  e = ( -p-yj  and the v a r i a t i o n a l terms u and v denote perturbations  from the reference point  n^) caused by p-j alone.  to case 9 i n Table 3-1 ( i n Section 3 . 6 . 2 , pages 3.21  This corresponds to 3.26). Equations  d e f i n i n g the motion of p^ were derived i n Section 3.4 (Equations and 3 . 3 4 ) , and f o r convenience these are  3.33  r e - s t a t e d below.  P o s i t i o n s of e q u i l i b r i u m at s p e c i f i c time i n s t a n t s can be defined by equating d e r i v a t i v e terms i n the equations of motion to z e r o ; a d i s c u s s i o n of t h i s topic and i t s i m p l i c a t i o n s i s given i n Appendix I I , q.v.  137 5.10  S  !  ^ — *  r  -  l  3  t- *"  I r  I *  When Equations 5.32 and 5.33 are s u b s t i t u t e d i n t o these expressions we obtain:  3  138 5.11 .where  S(K,V,  b)  =  f-i cost •+  f., t ^ . - O  (s--3s)  Note t h a t , because ( ? » n ) i s a f i x e d p o i n t , no d e r i v a t i v e s of ^ L  appear i n Equations 5.36 and 5.37. S(u,v,t)  and „ H  -i— ~  2  .  u n u  ~  We can evaluate the  expressions  and T ( u , v , t ) from Equations A1.6 and A1.7 i n Appendix I  pages A1.2 - A 1 . 4 ) .  J —  ~J  -  [-/^  »  Next, however, the nonlinear functions must be expanded i n terms of e.  3  y  J i v c n  l i t  and  L  r - , L ^ u u  . -u ^ i> i UII  / i j  T  •  i r \ i » O  also depend on the small parameter e.  i  x. _  / i -r •  i  n / i  i WT  _J  The p o s i t i o n s of u...«.  9  (see  x  i  UMUJC  . — v>  l , , x . v i u  u  This feature complicates  • O i  i o  the  process of expansion c o n s i d e r a b l y , and d e t a i l s of the a n a l y s i s are therefore relegated to Appendix V( Section A 5 . 1 ) . the f o l l o w i n g  |  k u..+ 0  Eventually we obtain  results:  k, j \ it. +  V/vfJ i - 2 . j O J  ^  C5-.40)  139 5.12  - £  O  -  CV)  e  Note here that eu and ev are both assumed to be 0(e ); the j u s t i f i c a t i o n f o r these values i s given on page 5.12 .  ^  11—- 2 V  -  «<- [  * -  ( k  0  + k, >  We now have  114.0  5.13  £ ^ V •+ ZU,  ILW^^CI- 2/0  +  -  and,  <f  '  l  -  by s u b s t i t u t i n g f o r £  L  O  -  +  and n  L  vy £ I  + 3k, ) j  O^fc ")  ^  ,  1  (5.43)  i n Equations 5.20 to 5.22, a t  and  these equations can be w r i t t e n a s :  e £  11 -  zv...+ 0,u-+  e £ v •+ zi.  Ar o^u.  O^v^  + o^v  J  =  ^  (i_ k  ^ 0 - k  =  In t h i s form the l i m i t process e  0  c  V  +  ^  0-44)  OCVO  o a O  ..  (5.45)  0 appears to be i n v a l i d , but the  constant terms i n v o l v i n g kg a r e , i n f a c t , dependent on e, as demonstrated below. When kg f 1 there w i l l be some e r r o r i n representing the functions  _L_  and  at L - and U (where eu = ev = 0 ) ,  1  which i s caused by the process of t r u n c a t i o n described on page 5.4..  It  i s not unreasonable to assume that t h i s e r r o r i s of the same order as the i n t e r v a l w i t h i n which the expansion i s v a l i d , and therefore i f the i n t e r v a l i s 0 ( e ) i t follows that 1  ( k - 1) = O f e ) . Q  1  A proof of t h i s  conjecture f o r Chebyshev polynomial approximations i s given i n Appendix VI ( i n Section A 6 . 4 ) .  For the present problem the region of v a l i d i t y  i s chosen to contain the f i n a l o r b i t (see page 5 . 3 ) , so that the order c l a s s of. the truncation e r r o r w i l l depend on the leading term i n the  141 5.14 s o l u t i o n s f o r eu(t) and e v ( t ) .  These s o l u t i o n s depend, i n t u r n , on  q u a n t i t i e s i n Equations 5.40 and 5.41  that are independent of u and v  and of the lowest order i n c, namely e | 2 k ^ . ^ e^j and e J 2 k , p ^ J f j • s  s  i  I f e^ and f g generate no secular terms i n the s o l u t i o n s f o r eu(t) and e v ( t ) , we are therefore j u s t i f i e d i n assuming ( ) from Equations 5.44 and 5.45,  = O(e^) so t h a t ,  as e -* 0 the o r b i t described by eu(t)  and ev(t) w i l l contract to one of the e q u i l a t e r a l t r i a n g l e points  or  Note p a r t i c u l a r l y that the homogeneous s o l u t i o n s of Equations 5.44 and 5.45 contain only p e r i o d i c terms of the form cos(to t + 0) and s  cos(co t + <j)).  From Equations A4.19, A 4 . 2 1 , A4.28 and A4.29 we have:  +  L  o  ^  ^  I + k  *•  (s.*o  + W," - 3 k % ( i - / 0 ] V  <*.*?0  + k, -+ [ -vdc^w,) + k * -  k  e  +k  (  - [ 4(k  and consequently, since k  Q  0+  0  and k-. are  2  ^^c\-fA )] Z  (  constant: j..  Appendix IV, Section A 4 . 1 .  iij.2 5.15 For a binomial expansion of the nonlinear f u n c t i o n [1 + p]  3  A  k  A  Q  - 1  3 and k-j. = y » i n which case ^  -  1  V  ^  °  (S.5o)  .  (5.50  where the e q u a l i t y condition applies when u -»- 1.  The change i n these  2  values of k  Q  and k-j v/hen a d i f f e r e n t expansion method i s employed w i l l  depend on the accuracy of t h i s f i n i t e expansion, but i n p r a c t i c e changes are s u f f i c i e n t l y small that Equations 5.50 and 5.51 used (see Appendix V I , Section A 6 . 5 ) . generated by e ( t ) 5  the  can s t i l l  I f no secular terms are to be  and f ( t ) , we must therefore require that y 5  2  f 1•'  When t h i s c o n d i t i o n i s s a t i s f i e d , the functions eu(t) and ev(t) can be expressed as asymptotic expansions i n the two t i m e - l i k e v a r i a b l e s t and T w i t h , as before, T = e t .  From Equations 5.32 and  5.33 we then have *— i  i~—»  •n. « 5  .  +  .  .  .  For the general two body s o l u t i o n i t i s assumed that neither mass i s i d e n t i c a l l y zero (see page 2.11 and the footnote to Equation 2.38 on page 2.18). The r e s t r i c t i o n u f 1 i s consequently i m p l i c i t i n t h i s entire analysis. 2  be  11+3 5.16 where these expansions are to be uniformly v a l i d i n t and T as e -»• 0. The formal operation of d i f f e r e n t i a t i o n i s again assumed to be j u s t i f i e d , and the d e r i v a t i v e s of Equations 5.52 and 5.53 are,, i n f a c t , i d e n t i c a l to the expressions defined i n Equations 4.5 to 4 . 8 . We can now expand Equations 5.44 and 5.45 as a sequence of nonhomogeneous p a r t i a l d i f f e r e n t i a l equations where, f o r the i ' t h power of e :  hi?-  1  J  Here the functions S  u.j and v ^ .  it*-  and T  involve u  to u . , and v to v . , , but not o l-l 5 i-l c  c  The f i r s t p a i r i n t h i s sequence are  *t  *"  l\ + . + o u. + o v = e" /y• Ci-iO ^ft) ts.S7) dl k It. and i f the homogeneous s o l u t i o n s of Equations 5.56 and 5.57 are denoted A  7  5  3  5  s  t  d  by u ( t , x ) 5  and v ( t , x )  H  5  ' i<-*F')fr  =  *  -  u  5  H  r e s p e c t i v e l y , then from Equations A4.30 and A4.31:  o-Ct?) o-s [ fc> t + 6<lv)^ s  o  s  a c e ) cc.5 [ <o fc + 5  9*ttO]  +  h(t)  +  ffl  cos  u  [ iO t + u  e/>Cr)J  (5.58)  U r ) c« ]_ m fc + < £ * ( - & ) ] u  (5.51)  where, from Equations A4.34 and A4.35:  &* Co)  -  9Cv)  +  2-3 y  (s.6o)  11*4  5.17 Homogeneous s o l u t i o n s of higher order can be accumulated i n t o and V g ( t , r )  H  i f the a r b i t r a r y functions a ( x ) ,  Ug(t,x)^  b ( x ) , 9(x) and <J>(x) are  expressed as asymptotic expansions of the form  L—.  z> Z  i  Z—-  (see page 4.6 f o r a comparable process i n the three-body c a s e ) .  Also,  by r e - l a b e l l i n g x as e t , the non-homogeneous s o l u t i o n s f o r u^ and v.. (where i > 5) can always be considered functions of t , and consequently no d e r i v a t i v e s of these q u a n t i t i e s with respect to x need appear i n the sequence of p a r t i a l d i f f e r e n t i a l Within an e r r o r O ( e ^ ) ,  equations.  Equations 5.38 and 5.39 can be evaluated  d i r e c t l y from Equations A1.6 and A1.7 by s u b s t i t u t i n g £^ and ri|_ i n place of £ and n . and i f we express these equations using the compact notation  SCV.v, t )  = \  S.CO &  Is  TtV,v,fc)  +  l  OtVO  b  = \  T. C t )  £  + OCV')  :  the sequence of p a r t i a l d i f f e r e n t i a l  equations can be w r i t t e n i n  form given below.  d\  s  + 2_5u  cJt -  5b  5t  5b  1  s  + 0 u. z  5  tOjV  s  e~ rf(i-k }  «.  0  %\Y^"I  •  e Ct) s  L  Ct)  5 l  -  S  -  - 2-  <3tatr T CO 6  -  5v __5 OC.)  Z k f ^ e / t }  - 75t5V  •it*  5b  3V  5.19 x  C O  T  5"?  f  -  a i c , ^ ^  t ?  f t )  ?  •  aV  - 2-  5  L atar  -  •4- J, a  + oy  ?  l  9  at  at 1  - 2  - 2>J  aV-  at  +o a^ + 0 v  g  x  3  g  'oC.)  -  +  - ^ \  J  z>a  CO  _  $  TV  +  -  av  -  T  It -  CO  at J  1  oCt)  -  Z W , ^e  s  ( 0  at  -  2-  i-  OCe )  a, -v p V5 -t- ciu.5  ^Ct+jO' 1  OCi)  06O  v  ^_«-<} ...-.2 a_Vq + 0 ^ + O^V^ a if  s/t)  -  ^ . ^ A ^ C O  fc  -I r  ^ atr  . aTai  av . v  - ay at  h\  at 1  4  aaa, +  0 ^  a, +  =  + za u x  t  atar  T^CO -  ^  at  e  ] J  (  </0  (???)  u  2- ^"Vj  -  au.  5  0( O  j 1 ae-  + ^.5 7  -»- aj*  -f  at J o 6 ^ e  L  t  at 1  at  a%/ " it; 5  1  £  1  - av a* s  s  + 2a\  s  ik^V  ^w.^ ? avj  114-7 5.20  c>fc  - 2-  a\i  1  -  5  ar  1 bv-  at 1 c , ^  L it*-  0  bt  L bt^  u e 5  5  (3k, + k^)  a>  ^tatr  bt  *\  —  •  J OCe~)>  bv  at*- J > , oCl  1- V ^ Ck,+3k,.) 5  s  o(0  at - 7-  at*  oCe*)  aw-  ^  oCO  J o  oCe ) 3  1i+8 5.21  J  As before (page 4.8)  the s u b s c r i p t 0(e )  0CO  denotes terms of order e  1  in  1  the associated asymptotic expansions. -  5.4  Uniformly V a l i d Solutions The method of s o l u t i o n adopted here i s e s s e n t i a l l y i d e n t i c a l  to that employed i n Section 4.3 f o r the three-body problem, where a uniformly v a l i d s o l u t i o n i s determined by e l i m i n a t i n g a l l secular or unbounded terms.  Because the a n a l y s i s becomes both p r o l i x  and  r e p e t i t i o u s , only Equations 5.68 to 5.71 are considered i n d e t a i l , although the e n t i r e development i s given i n Appendix V, Section A 5 . 2 . The non-homogeneous functions i n Equations 5.68 to 5.79  can,  i n g e n e r a l , be w r i t t e n i n the f o l l o w i n g form:  • —I  r  ( 5 . 8c») '4  /  1  J  11+9 5.22 -where E  and F  are defined i n Appendix I V , Equations A4.J36 and A4..37.  Corresponding non-homogeneous s o l u t i o n s are given i n Section A 4 . 2 , and these f u r n i s h the required conditions f o r the e l i m i n a t i o n of secular terms. For Equations 5.68 and 5.69, and f ( t )  from Equations 4.101  K  and a f t e r s u b s t i t u t i n g f o r e,-(t)  and 4.102, we have:  A" (  ' F* Cb,X)  =  fe"  ^ C\- k ) D  +  i  u C i  HlJ  III  oi  i*  »v-  V-^MU  ^  « W I I ^»  ***.v-  + |A, ) i  ^ K l f i f t d - ^ 8  A  I  . _ _  . < v  •  (sVs'i)  O + | 0  .  .  . .  >  "I  "'  w  * "  '  •*'  equations A4.40 and A4.41, the corresponding non-homogeneous s o l u t i o n s u (t) 5  N H  and v ( t ) 5  N H  can be obtained from Equations A4.44 and A4.45 as:  150 5.23  -S  *^.5 Si.«vt 4- O ^ Q o b  L (S-.84)  '  5  {  O - O  {  )co t S  o  u  C u> * - t J ^ ) C o J - 5  I )  In these s o l u t i o n s f o r U g ( t ) ^ and v ^ ( t )  NH  terms associated with  can be obtained d i r e c t l y from the corresponding short period (co ) terms s  2 if u  s  2  i s replaced by  (note, however, that (u)  s  2 + co^) becomes  2  - ( u > - w ) a f t e r the interchange). s  L  A s i m i l a r instance of t h i s type of  symmetry i s mentioned i n Appendix IV, page A4.13. We next consider non-homogeneous s o l u t i o n s of Equations 5.70 and 5.71 f o r U g ( t , r ) and v ( t , T ) . g  The functions S ( t ) g  and T ( t ) g  can  151 5.24 be evaluated from Equations A1.6 and A 1 . 7 , and from Equations 4.101 and 4.102:  e Cb) 6  *s (b) b  *  p., C i - j O £  n. s» z t  I t therefore f o l l o w s that  (5.S8>  hilt-  Sin.Zb  - S:~2.b  {  +  1^1  toZb  The expansion of d e r i v a t i v e terms using Equations 5.62 to 5.65 i s a straightforward but lengthy o p e r a t i o n , and f o r t h i s reason i s relegated to Appendix V (Section A 5 . 2 ) .  When the d e r i v a t i v e  expressions  i n Equations 5.70 and 5.71 are accumulated using Equations A5.50 to A5.70 we o b t a i n :  152 5.25  5tr  L at  ire  '  av  J  t  ar  3  av  (s\ qo) t  I  J  ar  ar  2-  -.2.  J  5a  0  Qo C w b -f s  av  ae>  0  at  av  (see page A5.15).  and  at;  ar  153 5.26  (5.1') Secular terms i n the non-homogeneous s o l u t i o n s f o r Ug and Vg w i l l be generated by a l l the expressions i n Equations 5.90 and 5.91 involve p a r t i a l d e r i v a t i v e s with respect to x.  If,  that  f o r the moment, we  neglect the remaining q u a n t i t i e s , then Equations 5.90 and 5.91 can be written:  2 co.  + COS (J t L  +• COS O j  t  151+ 5.27  •+• c o s c O t u  (S.<?3)  The symmetry between short and long-period terms, already noted on pages 5.23 and A4.13, i s p a r t i c u l a r l y apparent here.  I t i s only  necessary, t h e r e f o r e , to consider s h o r t - p e r i o d terms i n the f o l l o w i n g a n a l y s i s , as corresponding r e s u l t s f o r the l o n g - p e r i o d terms can be obtained by i n s p e c t i o n .  I f we now take  av  av  I s  6  - zo  5, =  5  o>  J  atr  5  -S  av  av  L atr  p  L av  0  o  yc  J  L  s^0 * + A 39„ <*>8*? - 2.5" ^o. o  o  L av  0  av  j  t aatv  J  av  c ^ 0 / - a a £ , ^ e / ? •+ 2-5" J ^ s ^ B , +  f  | ^  L  <feo  .a^  9 7 c>  J  ar  then, from Equations A4.50 and A 4 . 5 1 , secular terms i n v o l v i n g u> 'are s  eliminated from u and v when: g  g  155 5.28  A f t e r s u b s t i t u t i n g f o r YQ» Y-J » <$  and 6-j, Equations 5.98 to  0  101 can be expressed as:  +- a,  = o  = o•  =  ftptoC  ^ Z.^^^" -0-4^0^ 1  + « tt^$ ^ 9  0  2(«  - o Now from Equation A4.34:  (if. io«.)  o  5  l  -0  - ^ s ' j  (£. las')  156 5.29  0  6  T  2.60  arctcxiv- f  so that we have  =  si* 0  Cos  B  =  sc^,  9  cos  9  COS 9 $  +  to* 4  ZcJ  5  4 ^ ) ^ -  S  in.fii  2c0 '  A f t e r s u b s t i t u t i n g f o r the corresponding values of sinGg Equations 5.102  (5.  s  and cos6g  to 5.104 can be w r i t t e n i n the f o l l o w i n g form:  2  -  O  1 :  2  C  3  i  «-o  CL  0  0  o  2_  where c-|, C2> c^ and c^ are constant q u a n t i t i e s i n v o l v i n g Q , to , s  and v-,.  s  , V2  'o?)  157 5.30 The only possible s o l u t i o n of these equations i s  JL  (  Q-o  ")  =  O  and consequently we obtain  A l s o , from the symmetry of Equations 5.92 and 5.93:  A l l the d e r i v a t i v e terms i n Equations 5.90 and 5.91 f o r E*( t,x) and F * ( t , x ) are therefore e l i m i n a t e d , so that we have  (5.117-")  158 5.31  F*£lO  -  +  Cos  Zb  The non-homogeneous s o l u t i o n s u ( t ) g  and v ( t ) can now be derived g  from Equations A4.40 to A4.45 ( i n Appendix IV, Section A4.2.1)  a  -  lb)  V>  j * , cos cJ t s  r  S  - 1°*-^  KM«-JO+J. ^  as:  159 5.32 jK Cos e«?i.b  +  t  +  '  3_ 2.  V Cfc) 6  4  -  p  x  =  -  ^  si-_ to b s  cos o  s  k  (  [«oJ-+ 4 - - o ] 3  +  I"  1 ]  160 5.33  f*, Si*.Zt  4  . ( ^ - 4 X ^ - 0  - p j cos Z t The f o l l o w i n g r e s u l t s are condensed from the continuation of t h i s a n a l y s i s , which i s given i n Appendix V as f a r as s o l u t i o n s f o r Equations 5.78 and 5.79.  air  at =  ^ 3  av  air  4k  =  av  a_p  3  =  constant  O  161 5.34 =  constant  (5.127)  =  constant  CS.\T3)  =  constant .  (s.af)  I t i s not u n r e a l i s t i c to set k  Q  = 1 (see Appendix V I , Section  A6.5), and i n t h i s case the s o l u t i o n s f o r Ug(t) and v,-(t) reduce to the form given below:  (5.134)  162 5.35 The s i g n i f i c a n t reduction i n U g ( t ) ^ and v ^ t ) ^ (Equations 5.84 and 5.85) r e s u l t s from s p e c i f i c values of  and  which cancel  short and long-period terms i n the non-homogeneous s o l u t i o n s .  For a^  and bg we have:  C4o9 t^s"  The values of a . and b. depend on r e s p e c t i v e l y (see page 4.20  1  0  -CiO,  Sl«.0  a n c  ' ^2"  7  (ffSS)  - O  ^J&Ui at?  and  ua. at  f o r an equivalent d i f f i c u l t y i n the three-  body c a s e ) , which prevent the rigorous determination of a-j, a^, 6-| > 02' *h  D  b-j, b^,  However, from the general trend d i s p l a y e d by  |^ and jj^ i n equations 5.115, 5.117, 5.124 and 5.125, we might well be j u s t i f i e d i n assuming that constant terms i n Equations 5.126 to 5.129 are z e r o .  a,  When t h i s c o n d i t i o n i s s a t i s f i e d :  I  P  5  c^e  o  -  iO A s  s  9  e  ^  (^13?)  163 5.36  ( * . « 5 l )  8,  -  9,  L  -  ]  (  ,4  t0  5  A  s  sCvv ©  t  1  142.)  143)  where:  rs  161+ 5.37  +  6 2,  (S.lSo)  *3  (s.isz)  and  ( i = 1,2,  ••• 4). corresponds to  Equations 5.149 to 5.152.  when w i s replaced by s  in  165 5.38 Considering only the s h o r t - p e r i o d terms i n u  g  and v ^ , from  Appendix V, Section A5.2 we. have:  H  5  5  which can be expanded as:  CoS  (5.1 S 3 )  H  When  3  30j av  and  d j ^ "are z e r o , from Equations A5.245 and A5.246 1  h-e  (Appendix V, page A5.69) both  - a  a  9?  j  and  ja.Q,  +a ej_J o  166 5.39 are z e r o , so that terms 0(e ) can be eliminated from Equations 5.153 and 5.154. • A f t e r s u b s t i t u t i n g f o r a^ and a 8 ^ from Equations 5.135, 5.137 Q  and 5.141, u ( t ) 5  H  +  and v ( t ) 5  can be w r i t t e n i n the form:  H  oC*fc*)  ....  (s.i s o  When the homogeneous and non-homogeneous s o l u t i o n s are combined, the terms aQCOs(w t + 6 ) , s  Q  Q aQCOs(w t + 9^*) and corresponding long s  s  period terms are eliminated from u^(t) and v ^ ( t ) (see Equations 5.133, 5.134 and Appendix V , pages A5.42 to A5.45).  A similar cancellation  might be a n t i c i p a t e d between components of the s o l u t i o n s f o r u ( t ) , g  Vg(t) and corresponding" 0(e) terms i n Equations 5.155 and 5.156. The functions u ( t ) and v ( t ) are defined i n Equations 5.120 and 5.121, g  g  but i n more compact n o t a t i o n :  16?  5.40  52 - * * u  0  u  .  •2  v Ct) 6  *J C « U  s  l  - co^)  J  ,i  ^  «i.  .  / i  .  N  _  168 5.41  Although a proof i s not given here, the c a n c e l l a t i o n a f f e c t s a l l short and long-period terms i n Equations 5.157 and 5.158. Solutions of Equations 5.34 and 5.35, which we are now i n a p o s i t i o n to accumulate, are given below.  (sr. 151)  l  * ' 7 ./*</ ^'-/0 k  /  /  t  - €  I  3«Vs:».fc  cosb  169 5.42  -  £ /<-, co$ Z t &  5.5  Discussion Equations 5.159 and 5.160 describe the motion of  i n the  8 v i c i n i t y of  and Lg w i t h i n a constant e r r o r 0(e ), where the s o l u t i o n  i s uniformly v a l i d f o r t = 0(e~^).  As the small parameter e tends  to zero these o r b i t s contract to the Lagrange points L^ or L g , so that we may consider each e q u i l a t e r a l t r i a n g l e point as the l i m i t i n g member of a f a m i l y of four-body o r b i t s .  I t should be emphasized that  these unique s o l u t i o n s do not account f o r any perturbations other than those r e s u l t i n g , from the i n f l u e n c e of p-j.  The question of s t a b i l i t y  cannot, f o r example, be resolved without an a n a l y s i s of the more complicated s i t u a t i o n described as Case 10 i n Table 3 - 1 . ^  Note,  however, that Equations 5.159 and 5.160 both define the l i m i t i n g members of a f a m i l y of o r b i t s depending on the parameter 6 and provide the generating o r b i t s f o r Case 10. Section 3 . 6 . 2 , q . v .  We may deduce from the general  170 5.4 character of Equations 5.159  and 5.160  that p e r i o d i c terms i n v o l v i n g  s i n n t , cos nt are associated with the c o e f f i c i e n t  ! (o.'-rvSC^-n. ) 1  For n > 2 t h i s expression can be approximated by -^r ,  and consequently  higher-frequency terms w i l l exert comparatively l i t t l e influence on s o l u t i o n behaviour.  171 6.1  6.  6.1  SPECIFIC SOLUTIONS  Introduction The s o l u t i o n s derived i n Chapters 4 and 5 f o r three and f o u r -  body motion permit, w i t h i n the l i m i t a t i o n s of the primary model, a comp l e t e l y general choice of values f o r between p^ and the mass-centre of  , y,, and a n  d  r^,  the  distance  Three p a r t i c u l a r cases  are considered here, two of which r e l a t e to three and four-body motion f o r the sun-earth-moon c o n f i g u r a t i o n of our s o l a r system.  The remaining  three-body problem was selected to i l l u s t r a t e the accuracy with which unusual o r b i t s can be p r e d i c t e d , and a p p l i e s more to s t e l l a r than to planetary motion,  in each case the o r b i t i s p e r i o d i c , and a standard  of comparison may be obtained from a numerically-generated o r b i t having the same p e r i o d .  6.2  Three-body Orbits The s o l u t i o n s defined by Equations 4.101  to 4.104  are ordered  i n terms of e, but when computing a s p e c i f i c o r b i t i t i s more convenient to rearrange these expressions £  (b)  =  Ii-  y.^  i n the f o l l o w i n g form:  - fe1, p / i - f O  -  pJ^Vx.)  1,  2.  Cos t + . 10S e^pt, ( I- j / O l ? S ( I + p, )  +  l*\  ~l  172 6.2 -cc.s2t  • JL "VI ('-/-O cos k'e 6  £4-  (o.l)  € ( H-J*  r- Sin. 2-fc  S  12.  L 33-  63  3^  e'  0  -^z.")  I0Z40+/O J J  r:*u4t  256 .(.t.O  If ( t ) 3 : p  cos b  =  -  Pa.  ' ^ V ^ i  +  £ 6  */z  ( |+ M i ) .  0 - f O L  5  z  3zd\+fO ^  173 6.3  + cos 2 k  64-  32.  (6.3)  8  12-  L 3z  10x4.(14^ ,  ) J J  2.5-6  32.  We need only consider the o r b i t of one p a r t i c l e , ' and i n t h i s  section  a t t e n t i o n i s therefore confined to the motion of p 2  Equations 6.1 and 6.2 d e s c r i b e , with an accuracy O(e^), an o r b i t which i s p e r i o d i c i n t = 2TT. With t h i s as an i n i t i a l 'cf.  Equations 3.5 and 3.6.  estimate,  6.4 .we .should be .able to f i n d p e r i o d i c s o l u t i o n s of the o r i g i n a l equations motion (Equations 3.31  and 3.32)  using a numerical approach.  It  of  is  important, however, that the computed o r b i t be p e r i o d i c i n t = 2TT and not t = 2TT -i- 6, where 6 i s some small q u a n t i t y .  One method s a t i s f y i n g  t h i s c r i t e r i o n has 'been o u t l i n e d by Bennett and Palmore [59 ] , f o r which the t h e o r e t i c a l  6.2.1  basis i s given below.  P e r i o d i c solutions of y ( t ) = B(t)y +  f(t)  I f X(t) i s a fundamental matrix s o l u t i o n of the equation  x  =  A(t)X  A X '(t)  = - X~'(t)  (6.5)  fl(t)  (6.6)  Now suppose  where Y(t) i s a fundamental matrix s o l u t i o n of  U.8)  175 6.5 We then have, from Equation  j u. L  6.6:  -*  tit'  and t h e r e f o r e , i f t  = 0:  Q  j  Y~ Cb)y(b)= ]  1~'LX)$CZ)AX  +  constant.  (6.io)  o I f Y(t) i s the p r i n c i p a l matrix s o l u t i o n , Y(0) = I (the i d e n t i t y m a t r i x ) , so that  +|  lb) c^Co)  We now assume that the l i n e a r system y = B(t)y has s o l u t i o n of period T except the t r i v i a l s o l u t i o n y = 0. i f f(t)  is T-periodic,^  Equation 6.7  no p e r i o d i c  In t h i s  case  there e x i s t s a unique T - p e r i o d i c s o l u t i o n of  [60 ]. 2  I f Equation 6.11  i s to be T - p e r i o d i c ,  then:  T ycT) (o') 4  -f f y c T )  T h i s contraction period t = T.  +  y~ c-v) icx) AX l  =  ^  to')  c&.>o  i s used to denote that a f u n c t i o n i s p e r i o d i c with a ,  176 6.6 which defines y(0)  Co)  =  as:  [  y-'Cr)  -i  - l ]  r  y.-'CuKctOAV . .  c&  . ) l3  The T - p e r i o d i c s o l u t i o n can then be w r i t t e n i n the form t-t-T  Now suppose that p(t)  i s the s o l u t i o n of Equation 6.7 s a t i s f y i n g  i n i t i a l c o n d i t i o n p(0) = 0.  In t h i s case [61  the  ]:  t p(t)  -  j" YCt) W  r  and t h e r e f o r e , from Equation  tjCt)  I f Equation 6.16  = YCtOyCo)  )  K r )  ditr  <6.»0  6.11:  + j=(t)  i s to be T - p e r i o d i c :  .  '  (6.ifc)  177 6.7 YCT)  3 CO) + pCT)  = yCo*)  Cfe-i?)  so that  3co)  =• - [ v c r ) - x ]  pCT) .  a.is)  We now have s u f f i c i e n t information to generate a T - p e r i o d i c . solution i t e r a t i v e l y .  The algorithm given i n the next s e c t i o n  is  s i m i l a r to that described by Bennett and Palmore, although they propose a d i f f e r e n t updating c o n d i t i o n .  The proof of convergence appearing  i n [59 ] does, however, apply to both algorithms.  6.2.2  I t e r a t i v e method to determine a T - p e r i o d i c o r b i t The system equations are w r i t t e n i n the general form  iCt)  =  ^ [ xCt") , t]  U.n)  and we expand the function g [ x ( t ) , t ] about a T - p e r i o d i c estimate to o b t a i n :  o-vc  x (t) k  178 6.8 Setting x = x ^ , + 1  Equation 6.19  where the terms 0(|| *  w +  , - *  can be expressed as  II*)  k  are neglected.  ing a s s o c i a t i o n s between Equations 6.21  H (t)  x =  ^9  and  VJe now make the f o l l o w -  6.7:  Ct)  • U.ix.)  [* .  (6.as)  =c .'  k  k+|  ax fCt)  s  .-3[=c .tJ  " ^ [ *  k  k  J  t ] *  0 * 0  k  2x  and require that the s o l u t i o n x ^ ^ ( t ) of Equation 6.21 +  be T - p e r i o d i c .  The algorithm to determine x ^ ^ ( t ) i s summarized below. +  1.  Integrate Equation 6.21  over the i n t e r v a l 0 ^ t ^ T f o r the  s o l u t i o n estimate x^(t)  with zero i n i t i a l  c o n d i t i o n s , and  denote the r e s u l t by p ( t ) . 2.  Integrate the matrix  vct)  « . ^ [ x  equation  k  l  t ]  yet)  C6.zs)  ax over the i n t e r v a l 0 s= t ^ T with Y(0) = I (the matrix).  identity  This generates the p r i n c i p a l matrix s o l u t i o n of  179 6.9 equation 6.21 when the terms  <j [ x , k  t] - ^ t  3 C  k  . ] t  a  k  a  r  e  neglected. 3.  Determine x -|(0) from Equation 6.18.  4.  Determine x^ -|(t) from Equation 6.16.  5.  Set x^(t)  k+  +  = x ^ - j C t ) and i t e r a t e from step 1 u n t i l  satisfactory  convergence i s obtained. The matrix [Y(T) - I] can be almost s i n g u l a r , and i t i s p o s s i b l e , because of accumulated e r r o r , f o r the i n v e r s i o n process to destroy convergence.  In t h i s case the t h i r d step of the algorithm may need some  m o d i f i c a t i o n to avoid inaccurate elements i n the c a l c u l a t e d vector x  k + 1  (0).  6.2.3  +  P e r i o d i c earth o r b i t For the sun-earth-moon c o n f i g u r a t i o n of our s o l a r system the  constant ft i s determined by the r a t i o of the s i d e r e a l mean motions of the sun and moon as:  ft  =  7.4801  32855 x 10~  2  .  Note the s l i g h t d i f f e r e n c e between t h i s r e s u l t and that quoted f o r ft on page 4.36  which a r i s e s from the use of current values f o r the mean  This d i f f i c u l t y was encountered i n d e r i v i n g the. three-body o r b i t s of Sections 6.2.3 and 6 . 2 . 4 . The two elements adversely a f f e c t e d by the matrix i n v e r s i o n were, i n i t i a l l y , both z e r o , and convergence was restored by r e t a i n i n g these values i n subsequent i t e r a t i o n s .  180 6.10 motions (given i n the footnote to page 4.34).  We then have, from  Equation 3.21:  y  1  =  3.2979  46384 x 10  5  No a l t e r a t i o n i s necessary to the numerical values of r-j and ( l - y ) 2  given on pages 4.37 and 4.46 r e s p e c t i v e l y , so that r  1  =  3.8917  y  2  =  0.98784  l-y  2  =  1.2150  24558 x 10  2  96860 31396 x 1 0  - 2  The corresponding o r b i t f o r the earth can now be obtained from Equations 6.1 and 6.2 a s :  5  C t "i  =  1.7-132  +• %.X iiO £  <?2334  63^34-  *• lo"" " 1  x lo~'°  Ccr^3t  181  6.11  + i.  • -  *  oi~?2-6 v (©"^ s;~zfc  1 . f^2-  f&joo  sin  s xo ° X  3fc  83o<?4 mo'^siiv 4 t  1-630?  .  This s o l u t i o n i s shown i n Figure 6-1 i n terms of the displacements A£ and An from the point £ = l - ^ ,  n = 0.  Equations 6.26 and 6.27 were  evaluated at the time instants t = 2nir/100 (n = 0 , 1 , 2 , * • • ,100), and the data points of Figure 6-1 therefore define the p o s i t i o n of p^  a  ^  successive i n s t a n t s of time separated by At = 2IT/100. To v e r i f y t h i s s o l u t i o n the o r i g i n a l equations of three-body motion (Equations 3.31 and 3.32)  were solved f o r a 2ir-periodic o r b i t  using the algorithm described i n Section 6.2.2  with Equations  6.26  t  and 6.27 as the i n i t i a l estimate.  When Equations 6.21 and 6.25 were  evaluated at 201 equally-spaced time i n s t a n t s 0 ^ t < 2TT the quantity  |x  k+1  ( 0 ) - x (0)|| k  over the i n t e r v a l reduced below 1 0 "  f i v e i t e r a t i o n s , i n d i c a t i n g rapid convergence. [62 ] 1  13  in  A Fourier i n t e r p o l a t i o n  of the r e s u l t i n g p e r i o d i c s o l u t i o n generated the f o l l o w i n g  expressions f o r  and  r^Ct):  A l l the numerical r e s u l t s presented here were obtained i n doublep r e c i s i o n on an IBM 370 Model 168 computer, using a Runge-Kutta i n t e g r a t i o n routine with v a r i a b l e step s i z e and e r r o r c o n t r o l .  •  • •  0.8 -  •  O.t*  1  •  -  1  I - o.t*.  -0.2  0.4-0-4 -  •  •  •  - 0.8 -  •  •  • •  Figure'6-1 •  P e r i o d i c earth o r b i t about  £  9  = 0.012150, n  9  = 0.0  183 6.13  - 8 . 7 l t o  362-31  + 'I . 12-04-  (fc)  =  q.  io$\  * I O " " ^ cos  Z 2 0Z0  XI42,7  co$ 3 t  x 10  .* i o "  .  q  - 8  _  •V 1 . ^ 3 ^ 5  bb<\o">  -X.H10S  2-42-7^  A.  * io  * <o"\  -8  n^Zt 5C^3t  + 6 . & o 3 7 Soifcl  * io  a*. 4 t  I.H6£> ? j 5 ' 4 3  y; i o  S i * . £fc  -  c  ,.. .  (6.3.1)  -9 where terms l e s s than 10  i n magnitude have been  neglected.  We now assume that Equations 6.28 and 6.29 represent an exact p e r i o d i c s o l u t i o n of the three-body equations of motion.  The  d i f f e r e n c e s 6 ? and 6n between t h i s o r b i t and the a n a l y t i c a l s o l u t i o n defined by Equations 6.26 and 6.27 are shown i n Figure 6-2 f o r the i n t e r v a l 0 ^ t < 2TT.  The dominant e r r o r s are associated with terms  i n v o l v i n g cos 2t and s i n 2 t ; by comparing c o e f f i c i e n t s  i n Equations  6.26 to 6.29 the percentage e r r o r of these terms i s found t o be 3.5256% and 4.4130% r e s p e c t i v e l y .  Both o r b i t s are p l o t t e d i n Figure  181+ IO  6.14 0<5  -I  -I  2.TT  0-3  -J  S/>^ x 0-S  io  S  -,  o-2S A  .TT  O-XS  -i  -OS  -I  Figure 6-2  ZTT  P o s i t i o n e r r o r s f o r the earth o r b i t over one period  t  Arjxio  ^  4  —  • •  \  \ . . \  >  /  \  /.  /.  0.8  N  •\  -  \.  • /  /  \ • \  /'•  \  /'•  0.4  •\  -  / •  10'  -0.4  0.8  Figure 6-3  0  0.4  Numerical and a n a l y t i c a l solutions about ^  0.8 =  0.012150,  -  0.0  186 6.16 6-3 f o r the f i r s t quarter p e r i o d , 0 ^ t ^ TT/2, to provide some v i s u a l comparison between the numerical and a n a l y t i c a l s o l u t i o n s .  As i n  Figure 6-1, d i s c r e t e data points along the i n d i v i d u a l o r b i t s are separated by a time i n t e r v a l At = 2TT/100.  6.2.4  P e r i o d i c o r b i t s f o r s t e l l a r motion In Section 4 . 5 a  s t e l l a r model was proposed i n which two s t a r s  forming a close binary o r b i t a r e l a t i v e l y d i s t a n t s t a r of comparable mass.  Configurations of t h i s type may be described by Equations  and 3.32 provided the c o n d i t i o n l / 2 r «  3.31  1 i s s a t i s f i e d , and i n t h i s  s e c t i o n we i n v e s t i g a t e the p a r t i c u l a r case defined ....by y^ = 10, ^  0-5  =  and r-j = 50. Thi?  Choice  Of  n a r a m p t p r s  Hp>$pr\/ps  ^nmp  rnirmpnt..  Foliations  3  4.101  >/T.  to 4.104 w i l l e x h i b i t rapid convergence i f the quantity e t 5 6 i s s m a l l ; a l s o , i f terms of order e and e i n the equations are to  be of comparable magnitude, ( l - y g ) -  •  i t f o l l o w s that e * 0.5/y^ " , and therefore 75  When y^ = ( l - ^ ) e y-|' ~ 3  /2  =  - 0.125/y^.  0-5, If  y-j = 10 t h i s l a s t c r i t e r i o n i s s a t i s f i e d when r^ = 50, which completes the set of  parameters.  We can now derive the corresponding p e r i o d i c o r b i t f o r p^ from Equations 6.1 and 6.2  S^(fcO  -  -  c f . page  4.32.  as:  6.66 9 0  4.. I S S o  mo?  * io~ * lo~  S  fc  c&t  187 6.17  -7-  />f  f t . ) ..  -  8-4"5oo  I ,  2?S  446i5  "A  I c " ^  + t.«43?5 x  v. i o ' sin. b S  S i r i 3 t  lo~'°  4-fc  .  (6.31)  The e r r o r incurred by neglecting terms O ( e ^ ) by the quantity e  u-j ,  i s l i k e l y to be dominated  which i n t h i s case w i l l be approximately  2 x 10" . 8  The looped o r b i t described by Equations 6.30 and 6.31 shown i n Figure 6-4, where A£ and An denote the respective  is  displacements  from % - 0.5 and r\ = 0; the d i s c r e t e data points define the l o c a t i o n of p  2  at successive time i n s t a n t s t = 2nTr/100, where n = 0 , 1 , 2 « * * ,100. A 2ir-periodic o r b i t , with Equations 6.30 and 6.31  as the  i n i t i a l estimate, was determined using the algorithm of Section 6 . 2 . 2 . Only two i t e r a t i o n s were needed to reduce the q u a n t i t y below 1 0 " ^ ,  11x^^(0) - x^(0) j|  and the f o l l o w i n g p e r i o d i c s o l u t i o n was obtained:  -  4 . i < ? 2 - 5  <»I4?5  •,*  to' cost s  188 6.18  0.8  H  0.4  H  - i •0.8  •0.4  0.4 0.4  H  -0.8  Figure 6-4  P e r i o d i c o r b i t f o r .u, = 10, u C = 0.5, n = 0.0 2  2  9  = 0.5,  r , = 50 about  A ^ x l O  4  189 6.19  -  S . $"34?  4o$zo  g.44«?-  =  + S.6g<J3  where terms l e s s than 1 0 ~  IU  |0  co5 4 b  (6.32)  6$||5 * io~. Si«.fc 5  IS3S5  bb$$o  .+.1.3316  *  * | o "  % io""''  ?  Swy.Zfc  s ^ 4 b  ,  (6.33)  i n magnitude have been omitted.  Differences between t h i s r e s u l t and the a n a l y t i c a l s o l u t i o n (which would not be detected on the s c a l e of Figure 6-4) i n Figure 6-5.  are shown  The dominant c o e f f i c i e n t e r r o r s i n Equations 6.30  and 6.31, 2.9968% and 0.2154%, are associated with s i n 2t and s i n t respectively.  Note, i n Equations 6.1 and 6 . 2 , that terms belonging  to the sequence  appear i n the c o e f f i c i e n t s of cos 2t and s i n 2 t , which are therefore the most s u s c e p t i b l e to e r r o r .  190 6.2C  0-4  H  t  -I IT  o-4  H  -0-8  H  0-2  0-1  H  i TT  o-i  t  2ir  H  0-2  Figure 6-5  P o s i t i o n errors f o r the s t e l l a r o r b i t over one period  191  6.3  Four-body Orbits Near  and L  Equations 5.159 and 5.160 define two 2iT-periodic o r b i t s , i n the v i c i n i t y of  and Lg r e s p e c t i v e l y , corresponding to s o l u t i o n s  f o r the motion of p and p which s a t i s f y Equations 4.101 to 4.104. 2  3  With, the s o l a r system constants given on page 6.10 we have, from Section A 4 . 1 :  %^  =  i. £674  -  -l.2t?4  at l _ ?o6£4-  4  at U 5  =. - 4 . S 7 2 4 -  96€6o  K. /o~'  S • 6&OZ.  £404°  * 10  54-04-0  *  1  * . - 8 . 6662  co~'- at L  c  b  Note that v  2  and  change i n sign but not magnitude at L g .  corresponding o r b i t near  The  of the earth-moon system can now be  determined from Equations 5.159 and 5.160 as:  192 6.22 gCt)  1  Ct)  r  4.8731  474^1  * 10 - i  "5.?<iZl  4  *  4  i . o68D  ^ o z l * io"  +  S". 6?o8  84-639.  -«- 4- £547  4ST63S  -  =  g.  -  65"2|  *  78033  IO  v.  <x>s t  6  io"  *io"  3  3  sc^Zb  *  U - » 0  co 2 b 5  io  -1>  2 . 1361  *?4o58 x. io  I . 8560  7411 3  2-2281  I«'j44  + 5". 6 8 3 0  sc«. t  sc*-b  fe  v. l o ~ Cost x io"  x  7oS4S  3  $i*,Zt  lo"-..cos2t  :  An equivalent o r b i t near Lg f o l l o w s d i r e c t l y from t h i s s o l u t i o n when terms i n d i c a t e d by an a s t e r i s k are changed i n s i g n . The o r i g i n a l equations of four-body motion (Equations 3.33 and 3.34) were solved f o r a 2ir-periodic o r b i t using the algorithm outl i n e d i n Section 6 . 2 . 2 , with Equations 6.34 and 6.35 d e f i n i n g the i n i t i a l estimate.  Three i t e r a t i o n s were s u f f i c i e n t to reduce the q u a n t i t y _ i p  ll k -|(°) x  +  _  x  |<(0)ll  below 10"  , and a f t e r a numerical F o u r i e r i n t e r -  p o l a t i o n the f o l l o w i n g r e s u l t s were obtained.  •  193 6.23  4~. 8 7 3 6  -  63i?6  K ID'  9 . 9 3 3 4 - . 6<i74o  * to  s c ~ t .  +  3.35VS  A-BbZ^ x I O -4 cos t -  + S- 63S2.  3oxo5  -3 * io  s^Zfc  4  4.Z493  734(2-  _3 * io c  4  2o647  x io  s;«.3t  -  2.. 7o4-3  "37909  K I D  —  1 . 4 8 23  O63ol  -<> * |o cos 4 b  6.o64o  + z.fc465  o i i o ?  -6 i o s;*_-4--t  x  oS  2-t  co5"3t  (6.36) 8.  87532. *. io"  -2.4-366  9I Z 3 3  J< i o  ->.././.v>j  J 6 6 T |  y . i O  4 3.6460  67Z74  v io  -1.364-0  .3SIST-  io  x  Su^. t  Sii*.  £  siOb . fc  sc^4t  4  7.4-9X3  6 4 4 2 - 4 * 'O  cost  +  i .  .B*0'Yt>  1C l O  Co5  4  4 . 4-436  80674  * »o  ~  2 . 6666  SS304 x i o  «  ct>s3t -6  c©s4t  C6-3?;  4 . 2 7 36  63| 76  n io" - s a "> ' s ^ t  S. 8 3 3 4  6SII4  5. b?*Z  -3 -bozoS t. \o ' n^.7Jt  6..064O  2.142-i-  x. 6 4 6 4  <??pio  +  - fc>  * io  K  i o "  $j^3t •  6  s;«-4-fc  -  3. 3 5 < S  4-87S8  ^lo^cej.t  x io~ co 2.k  +4.2-49 3  734.11  -  8864-2-*  Z . 704.3  .1.48X3  Z5"9?-9  3  \o~  x l o "  S  cbs 31  6  cos4fc (6.38)  19k  6.24 —i  1 * 7 - 4 3 6£>  -  - 2 .  ZZ-S*?  1 M 3 I  3S67I  *  ?t  I O " * c^b 5  i o "  + 3. 64-6C?  -6 £ > 6 o Z £ •< to  -  ?71 7 I  I . -3640  3  * 10'  s ^ i t  si*. 4 b  -  7 . A H 2-3  6 7 ? 7 2  ^ lo"  -4.4.436  ^0ZZ«  -< < o " c o S 3 f c  + 2.6^66  M7§Z.  - 5.  cosh  S  bb\ 8 6  * lo"  6  cos4fc  O3<0  Terms l e s s than 10"  have been omitted from these expressions.  The  symmetry evident i n Equations 6.34 and 6.35 f o r o r b i t s near L . and L 4  i s maintained here, although the agreement between c o e f f i c i e n t s always complete.  c  0  i s not  These s l i g h t i r r e g u l a r i t i e s a r e , however, more  l i k e l y to r e s u l t from the e f f e c t s o f q u a n t i z a t i o n noise and accumulated e r r o r i n the i n t e g r a t i o n process than from any non-uniform system behaviour. The a n a l y t i c a l and n u m e r i c a l l y - d e r i v e d o r b i t s near  and L  g  are shown i n Figures 6.6 and 6 . 7 , where A£ and An denote displacements from the e q u i l a t e r a l t r i a n g l e point ( i . e . ? - £^ =  A£, n -  = An)  and d i s c r e t e data points define the l o c a t i o n o f p^ at successive  instants  2TT  of time separated by At = J^Q .  Differences between the a n a l y t i c a l  and numerical s o l u t i o n s over one period are p l o t t e d i n Figures 6.8 and 6 . 9 , which show that the p r i n c i p a l source o f , e r r o r i s p e r i o d i c i n t = 2TT.  Note, at l e a s t i n the present case, that the p e r i o d i c o r b i t  defined by Equations 5.159 and 5.160 i s dominated by perturbations p e r i o d i c i n t = TT and d i r e c t l y associated with p-,. Terms p e r i o d i c i n  195 6.25 Am  <-  x 10  Analytical  t = o,TT, xtr  z  Figure 6-6  P e r i o d i c o r b i t s near L, i n the four-body problem  solution  196 6.26  t =3J  0.8  -  - 0 . 6  -J  .  |  t = 0,Tr,2TT  Analytical  Figure 6-7  P e r i o d i c o r b i t s near U i n the four-body problem  solution  197 x  0-4-  **  to"  1..-  TT  ATT  TT  .aTT  o-z  0-4  J  >7  O-l  J  0-4  J  Figure 6-8  P o s i t i o n errors f o r the o r b i t near  198 5£ O-br  6.28  x Io"  1  o.i .  0-4  H  rr  ZTT  Tf  —i ZTT  J  5*7 x io* 0-4  -i  0-3.  •o-Z  H  J  Figure 6-9  P o s i t i o n errors f o r the o r b i t near L  (  .  199  6.29 t = 2TT r e s u l t from the i n d i r e c t influence of p-j on the motion of p and pg, and are r e l a t i v e l y small i n magnitude.  Perturbations  2  associated  with p-j and p e r i o d i c i n t = 2TT w i l l , however, be encountered when higherorder terms are included i n the s o l u t i o n . I t would be misleading to assess accuracy of the o r b i t s described by Equations 6.34 and 6.35 by comparing c o e f f i c i e n t s with the numerical solutions.  If,  i n s t e a d , we express the p o s i t i o n e r r o r s <5£ and 6n i n  terms of. the maximum displacements  |6C|  *  0.048913  |A5|  i <5nI  ^  0.043673  An  1  6.4  a x  1  and [An|  max  »  then:  max  max  Discussion Although the numerical s o l u t i o n s described here are i n r e l a t i v e l y  close agreement with predicted r e s u l t s , we cannot be c e r t a i n that the numerical and a n a l y t i c a l o r b i t s are e q u i v a l e n t .  This uncertainty i s  a consequence of step 5 i n the algorithm of Section 6 . 2 . 2 , and can be explained i n the f o l l o w i n g way.  When the i n i t i a l conditions are modified  there i s no guarantee that the eventual o r b i t w i l l be unique i . e .  for  the three-body case we may obtain o r b i t s corresponding to case 4 i n Table 3 - 1 ,  rather than case 3.  Section 3 . 6 . 2 , q . v .  S i m i l a r l y o r b i t s i n the four-body  200 6.30 problem may correspond to case 10 instead of case 9.  No adequate  r e s o l u t i o n of t h i s d i f f i c u l t y i s p o s s i b l e without a d e t a i l e d a n a l y s i s of motion i n v o l v i n g more general  perturbations.  Comparatively few e x p l i c i t s o l u t i o n s of the three and four-body problems have appeared i n a form compatible with the r e s u l t s of t h i s chapter, which makes any f u r t h e r comparison d i f f i c u l t .  H i l l ' s solution  f o r lunar motion i n the r e s t r i c t e d problem i s p o s s i b l y the c l o s e s t equivalent to the o r b i t defined by Equations 6.26 and 6.27, case has already been discussed i n Section 4 . 4 .  but t h i s  An a n a l y s i s of p e r i o d i c  motion i n the general three-body problem by Moulton [63 ] a p p l i e s to o r b i t s described as case 3 i n Table 3-1; more r e c e n t l y A r e n s t o r f [64] derived corresponding r e s u l t s f o r case 4 o r b i t s .  has  Unfortunately neither  a n a l y s i s can r e a d i l y be compared with the present approach. The four-body s i t u a t i o n i s e q u a l l y problematic, with only one o r b i t showing s i m i l a r behaviour to the s o l u t i o n described by Equations 6.34 and 6.35. motion near  Kolenkiewicz and Carpenter [ 1 7 ]  investigated  of the earth-moon system f o r Huang's very r e s t r i c t e d  model, using a numerical perturbation scheme to determine an o r b i t with a period of one synodic month.  Although s i g n i f i c a n t l y l a r g e r i n  s i z e , the r e s u l t i n g o r b i t e x h i b i t s some of the features shown i n Figure 6-6.  Equations 5.159 and 5.160  i n d i c a t e that the motion of p  2  and pg exerts comparatively l i t t l e influence on the four-body o r b i t (as discussed on page 6.29 )which may account f o r the success of the very r e s t r i c t e d model i n t h i s instance. near  Three r e l a t e d s o l u t i o n s  should also be mentioned, although s t r u c t u r a l d i f f e r e n c e s  in  the methods of a n a l y s i s prevent any comparison with the present approach.  201 6.31 In a subsequent paper [ 3 0 ] Kolenkiewicz and Carpenter examined motion about  when the primary bodies move i n coplanar o r b i t s and  s a t i s f y the equations of three-body dynamics.  Two nearly i d e n t i c a l one-  month o r b i t s were found, with a phase d i f f e r e n c e of 180°; the d i s p l a c e ment from  i s comparable to the earth-moon d i s t a n c e , and i t appears  l i k e l y that these o r b i t s correspond to case 12 of Table 3-1.  Similar  r e s u l t s have been derived a n a l y t i c a l l y by Kamel and Breakwell [31 using the Von Zeipel technique.  Schechter [28] has considered  ], the  s t a b i l i t y of p e r i o d i c one-month o r b i t s f o r the three-dimensional case i n which the earth-moon o r b i t a l plane i s i n c l i n e d to the e c l i p t i c . stable p e r i o d i c o r b i t was found, smaller i n s i z e than the numerical s o l u t i o n described i n [30 ] and which corresponds to case 10 of Table 3-1.  A  202 7  7. ' CONCLUSION  Two fundamental problems of c e l e s t i a l mechanics have been considered i n t h i s a n a l y s i s : r e s t r i c t e d four-body problem.  the general three-body problem and a Although a number of c o n s t r a i n t s  are  imposed, no assumptions are made which could i n v a l i d a t e the f i n a l solution.  A consistent and r a t i o n a l approach to the a n a l y s i s of four  body systems has not p r e v i o u s l y been developed, and an attempt made here to remedy t h i s d e f i c i e n c y .  is  The basic s t r u c t u r e of the four  body problem described i n Chapter 3 should f a c i l i t a t e extension of the present work to include the e f f e c t of more general  perturbations.  In the p a r t i c u l a r three-body problem under i n v e s t i g a t i o n two masses, forming a close binary system, o r b i t a comparatively d i s t a n t mass.  A new a n a l y t i c a l s o l u t i o n of t h i s problem i s found i n terms  of a small parameter e, which i s r e l a t e d to the distance the binary system and the remaining mass.  separating  The asymptotic s o l u t i o n ,  which i s determined using the two-variable expansion procedure, i s _14 uniformly v a l i d as e  0 fortune i n t e r v a l s 0(e  a constant e r r o r O ( e ^ ) .  ) and accurate w i t h i n  Although many terms are l o s t i n the  process,  i t i s p o s s i b l e to reduce the s e r i e s s o l u t i o n to a form which can be compared with G.W. H i l l ' s r e s u l t f o r the lunar v a r i a t i o n o r b i t . This c o n s t i t u t e s  a severe t e s t , but comparatively close agreement i s  found between the two . o r b i t s .  The predicted s o l u t i o n i s p e r i o d i c ,  a feature that allows f u r t h e r comparison with numerically-generated orbits.  Two examples are considered, the f i r s t of which r e l a t e s to  203 7.2 the sun-earth-moon c o n f i g u r a t i o n of the s o l a r system.  The second  example a p p l i e s to a problem o f s t e l l a r motion where the three primary masses are i n the r a t i o 2 0 : 1 : 1 .  In both cases the numerical and  a n a l y t i c a l s o l u t i o n s show close agreement, with an e r r o r below 5% f o r the sun-earth-moon c o n f i g u r a t i o n and l e s s than 3% f o r the  stellar  system. The four-body problem is. derived from the three-body case by introducing a p a r t i c l e of n e g l i g i b l e mass i n t o the close binary system. Unique u n i f o r m l y - v a l i d s o l u t i o n s are found f o r motion near both e q u i l a t e r a l t r i a n g l e points of the binary system i n terms of the small parameter e, where the primaries move i n accordance with the u n i f o r m l y v a l i d three-body s o l u t i o n .  Accuracy, i n t h i s case, i s maintained Q  w i t h i n a constant e r r o r 0(e ), and the s o l u t i o n s are uniformly v a l i d as E + 0 f o r time i n t e r v a l s 0 ( e ~ ^ ) .  The predicted p e r i o d i c o r b i t s  are compared with corresponding numerical s o l u t i o n s f o r motion near and Lg of the earth-moon system.  O r b i t a l p o s i t i o n e r r o r s are  found to be l e s s than 5%, and i t appears l i k e l y that an extension of these r e s u l t s to the next order i n e would produce a s u b s t a n t i a l improvement i n accuracy. Further work on t h i s t o p i c could take a number of p o s s i b l e directions.  The s o l u t i o n s presented here may f a i r l y r e a d i l y be continued  to higher orders i n e, but an extension to the three-dimensional case would probably be of greater i n t e r e s t . e i t h e r development.  No d i f f i c u l t y i s forseen i n  B.eyond t h i s p o i n t , however, the process of  a n a l y s i s i s l i k e l y to become considerably more i n v o l v e d .  The s i t u a t i o n s  which r e s u l t from more general perturbations have been i n d i c a t e d  201; 7 i n Section 3 . 6 , and i t would be important to proceed from one degree of complexity to the next.  In t h i s context Table 3-1 should prove  helpful i n d e s c r i b i n g system s t r u c t u r e .  On a much more general l e v e l  the e n t i r e approach may be used i n the a n a l y s i s of non-integrable dynamic systems, p a r t i c u l a r l y when i t i s f e a s i b l e to decompose the problem into a number of s u b s i d i a r y cases.  205 Rl REFERENCES  1.  Szebehely, V . , Theory of O r b i t s . 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Astronomical I n s t i t u t e s of Czechoslovakia B u l l e t i n , V o l . 22 (1971), p. 72.  45.  A l l e n , C.W., Astrophysical Q u a n t i t i e s . U n i v e r s i t y of London, The Athlone Press, 1973 (3rd e d . ) .  46.  Nayfeh, A . H . , Perturbation Methods. W i l e y , 'New York, 1973. p p . 64-68 and 259-262, p p . 228-303, p p . 18-19, "pp. 228-232. J  2  3  47.  C o l e , J . D . , Perturbation Methods i n Applied Mathematics. B l a i s d e l l , Waltham, Massachusetts, 1968.  48.  Van der Corput, J . G . , Asymptotic Developments I . Fundamental Theorems of Asymptotics. Journal d'analyse Mathematique, V o l . IV (1954-56), p. 341.  49.  E r d e l y i , A . , Asymptotic Expansions. pp. 5-25.  Dover, New York, 1956,  209 R5 50.  C o l e , J . D . , and Kevorkian, J . , Uniformly v a l i d asymptotic a p p r o x i mations f o r c e r t a i n n o n - l i n e a r d i f f e r e n t i a l equations. In Nonlinear D i f f e r e n t i a l Equations and Nonlinear Mechanics ( J . P . L a S a l l e and S. Lefschetz e d s . ) , Academic Press, New York, 1963, p. 113.  51.  Kevorkian, J . , The two v a r i a b l e expansion procedure f o r the approximate s o l u t i o n of c e r t a i n nonlinear d i f f e r e n t i a l equations. In Lectures i n Applied Mathematics, V o l . 7, Space Mathematics, Part I I I , American Mathematical S o c i e t y , 1966, p. 206.  52.  H i l l , G.W., Researches i n the lunar theory. American Journal of Mathematics, V o l . 1 (1878). ' p . 5, 129, 245, p . 13, p p . 129-147 and 245-266, " p . 130. 2  3  53.  Brown, E.W., An Introductory T r e a t i s e on the Lunar Theory. U n i v e r s i t y Press, 1896.  54.  Hagihara, Y . , C e l e s t i a l Mechanics, V o l . I I , Part 2 , Perturbation Theory. MIT Press, Cambridge, Massachusetts, 1972, p. 805.  55.  Kevorkian, J . , Uniformly v a l i d asymptotic representation f o r a l l times of the motion of a s a t e l l i t e i n the v i c i n i t y of the smaller body i n the r e s t r i c t e d three-body problem. The Astronomical J o u r n a l , V o l . 67 (1962), p. 204.  56.  Clemence, G.M.,.The system of astronomical constants. In Annual Review of Astronomy and Astrophysics ( L . Goldberg, e d . ) , V o l . 3 , Annual Reviews, Palo A l t o , C a l i f o r n i a , 1965, p. 93.  57.  A s h , M . E . , Shapiro, I . I . , and Smith, W . B . , Astronomical constants and planetary ephemerides deduced from radar and o p t i c a l observations. The Astronomical J o u r n a l , V o l . 72 (1967), p. 338.  58.  E c k e r t , W . J . , and E c k e r t , D . A . , The l i t e r a l s o l u t i o n of the main problem of the lunar theory. The Astronomical J o u r n a l , V o l . 2 (1967), p. 1299.  59.  60.  Cambridge  Bennett, A . , and Palmore, J . , A mew method f o r c o n s t r u c t i n g p e r i o d i c • o r b i t s i n nonlinear dynamical systems. AIAA J o u r n a l , V o l . 7 (1969), p. 998. H a l e , J . K . , O s c i l l a t i o n s i n Nonlinear Systems. York, 1963. * p . 17, p p . 27-28.  McGraw H i l l , New  2  61.  Coddington, E . A . , and Levinson, N . , Theory of Ordinary D i f f e r e n t i a l Equations. McGraw H i l l , New York, 1955, pp. 74-75.  62.  Lanczos, C. Applied A n a l y s i s , P r e n t i c e - H a l l , New York, 1956. *pp. 229-235, p p . 438-507, p p . 455-457, " p p . 229-248, p . 454, p . 516. 2  5  6  3  210 R6 63.  Moulton, F . R . , A c l a s s of p e r i o d i c s o l u t i o n s of the problem of three bodies with appl4cation to the lunar theory. American Mathematical Society Transactions, V o l . 7 (1906), p. 537.  64.  A r e n s t o r f , R . F . , New p e r i o d i c s o l u t i o n s of the plane three-body problem corresponding to e l l i p t i c motion i n the lunar theory. Journal of D i f f e r e n t i a l Equations, V o l . 4 (1968), p. 202.  65.  Henrici, P., 1964.  66.  F l e t c h e r , R . , A new approach to v a r i a b l e metric algorithms. Computer J o u r n a l , V o l . 13 (1970), p. 317.  Elements of Numerical A n a l y s i s .  W i l e y , New York, The  211  SOME ASPECTS OF THREE AND FOUR-BODY DYNAMICS PART I I :  APPENDICES by  PETER G.D. BARKHAM  FACULTY OF APPLIED SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA September  1974  r  212 A ii  TABLE OF CONTENTS Appendix I. II.  Page EXPANSIONS IN TERMS OF e  Al.l  THE REFERENCE LOCUS CONCEPT  .  A2.1  Equations of Motion  A2.1  A2.2  Numerical Explorations  A2.4  A2.2.1  Intersections with the £ a x i s at t = 0  A2.2.2  A2.5  P o s i t i o n s of e q u i l i b r i u m when  tfO A2.3 III.  A2.14  Limitations  A2.16  SOLUTION OF THE THREE-BODY PROBLEM  A3.1  A3.1  The Homogeneous S o l u t i o n  A3.1  A3.2  The Non-homogeneous S o l u t i o n  A3.2  A3.3  Expansions of s i n ( t + 0 ) and cos(t+6)  A3.4  A3.5  Continuation of the Uniformly V a l i d Solution  IV.  A2.1  A3.6  VARIATIONAL ORBITS IN THE RESTRICTED PROBLEM  . . .  A4.1  .  A4.1  A4.1  The Homogeneous S o l u t i o n  A4.2  Non-homogeneous S o l u t i o n s  A4.7  A4.2.1 A4.2.2  Non-resonant case . . . . . . . . . Resonances i n v o l v i n g s h o r t period terms  A4.8  A4.2.3  Resonances i n v o l v i n g l o n g period terms  A4.10 A4.12  213 A iii Appendix V.  Page SOLUTION OF THE FOUR-BODY PROBLEM  A5.1  A5.1  Expansion of the Nonlinear Terms  A5.1  A5.2  Asymptotic Expansion of the D e r i v a t i v e Terms  A5.9  A5.3  Continuation of the Uniformly V a l i d Solutions  VI.  A5.16  CHEBYSHEV POLYNOMIALS  A6.1  A6.1  Introduction  A6.1  A6.2  Function Approximation Using S h i f t e d Chebyshev Polynomials  A6.2  A6.3  Coefficients  of T * ( y )  A6.4  A6.4  Approximate Expansions  A6.4  A6.5  Expansion of the Function | r . |  k  di  A6.9  A iv  LIST OF TABLES Table  Page  A2-1  Astronomical Constants  A2.9  A2-2  Parameter Values  A2.10  A6-1  Chebyshev and Binomial C o e f f i c i e n t s f o r the expansion of  f(x) = [ l + x ] "  3 / 2  A6.13  215 A v  LIST OF FIGURES Figure A2-1 ,A2-2  A2-3  A2-4  A2-5  A2-6  Page Primary c o n f i g u r a t i o n at t = 0 . .  A2.5  Locus near L-j  A2.16  A2-2-1  Polar representation  . .  A2.16  A2-2-2  £ component  • .  A2.17  A2-2-3  n component  Locus near L  A2.17 A2.18  2  A2-3-1  Polar representation  A2.18  A2-3-2  £ component  A2.19  A2-3-2  n component . . . .  A2.19  Locus near L  A2.20  3  A2-4-1  Polar representation  A2-4-2  £ component  A2.21  A2-4-3  n component  A2.21  Locus near L  . . .  A2.20  A2.22  4  A2-5-1  Polar representation  .  A2.22  A2-5-2  £ component . .  A2.23  A2-5-3  n component  A2.23  Locus'near.Lg  •  A2.24  A2-6-1  Polar representation  A2.24  A2-6-2  5 component  A2.25  A2-6-3  n component  . . . . . . .  A6-1  f(x) = [ l + x ] "  A6-2  Region of v a l i d i t y f o r the expansion of f ( x ) =• [l+x]-3/2 near L and L  A2.25 A6.10  3 / 2  4  g  A6.ll  216 Al.l APPENDIX I  EXPANSIONS IN TERMS OF e  Consider the two expressions  £  tot  +  j*, L%<-%)  CAM)  and  where £,n define the p o s i t i o n of p ^ .  From Equations 2.43 and 2.44  the  p o s i t i o n of p^ i n the £,n>£ system of reference i s given by  t  =  - r  cos t  where t i s the independent v a r i a b l e defined by Equation 3.30. therefore express the quantity  l>c,l  since  r  as  b -/^s^b) +  i  We can  5  +  1  Cfil.3)  217 A1.2 2 A binomial expansion of Equation A1.3 w i l l be v a l i d when  2  E, +n  2 «  r-j ,  and consequently the two q u a n t i t i e s defined by Equations A l . l and A1.2 can be developed i n terms of the small parameter - L .  The f i r s t few  terms of t h i s expansion a r e : J*l *oSt +  = £ , cost  - ^ t o s t l " J_ -  3. ( j c © s t - ^ $ l n . t )  CAi.r)  which demonstrates the c a n c e l l a t i o n of terms i n v o l v i n g and  3.15).  With  e = —  A  x  (see pages 3 . 1 2  (from Equation 3 . 2 8 ) , a f t e r some t e d i o u s ,  but s t r a i g h t f o r w a r d , algebraic manipulation we obtain the f o l l o w i n g expansions f o r Equations A l . l and A 1 . 2 .  In, IV 1 8  1  8  X  x  •  1  5  (A(.£>)  218 Al .3  { 1  /».  6  +£  it>  L  &oZfc  15  ^4-t  4-  - -  -  s«vb  - i o £ «~-3b 3Z-  Cot  <^3b  - 1>S  -  tfc  iz8  l r  u  l  c^Sfc  5  + 3|5 yi^5b ? 3?J  32-  64-  si~Sfc ?  ^ 5 4 4  ^4-  /  6  15-8  -  6 V«  £  i  J  V  - <•{ [" I - 3<*,2.b][ ^  3  »1* to  <L zb - tos It  <.4t  • -  219 A1.4 35 (^,3fc -5-2-  «C3fc  to^t  •315 1-2-  - 'JJhS  <C5fc  64  Gob  +  64-  Cj33tr  IfS  — 3*5  <3-S  Qrp 5"b  «*«  1  1-  oCe'O  When £>n define the p o s i t i o n of p and 2  Equations A1.6 and A1.7 take the form  J  L  6  io  /  L*  8  s.3  ^/'-/O  It  7£-  4-  cosb  <6  +  Ijf5  cos  3b + 3T? cos  5k  ( A  I . * )  220 A1.5  it  — •  5  fc'^O-fO  2 a  5  9 c o s t +• if c e ^ t 1  j  -  - V  -*  3 si^fc + ] S St«.3t  $  L-  oCO  J  13 +  /i'  6  Oct)  3^zt)  - 3v< siM.2-tr  3_ si«.b  Xb  8  it  iS_ s;«.fc •+- w £  VS_ «;^3fc «  ^ S  +  s  0 - 3 « o 2 f c )  - 3 u . S^aZb^ s  2-  s^'St +• 3iS  64-  oft)  (A l . l )  Si«v £ fc  x-xe  it *  Hi X.  e  t'  3  11,6-fO  cxTfe)  £  1  5."-5 [  3. «.~b  L *  + is  • *  $^3b  .  -v/  s  3_ Q r o b - _1S < t e 3 t  d ©CO  221 Al 15 h 6  Ji  1  1- 3coZt~) - 3M.  where the subscripts 0(1),  0(e) and 0(e ) denote terms of t h i s order i n  the asymptotic expansions of u^ and  (i = 5 , 6 , 7 , « « « ) .  222 2.1  A  APPENDIX II  A2.1  THE REFERENCE LOCUS CONCEPT " 1  Equations of Motion The a n a l y s i s of three and four-body motion given i n Appendices  I I I and V was preceded by a l e s s - f r u i t f u l i n v e s t i g a t i o n , but the r e s u l t s of t h i s approach are summarized here to i n d i c a t e the l i m i t a t i o n s which were encountered.  These two paths of i n q u i r y s t a r t to diverge at  Equations 2.47 and 2.48, i n the  ri plane.  which describe the motion of p  4  near p  2  and p  3  We have, f o r the planar motion of p^ :  I— 1 Lagrange configurations i n the r e s t r i c t e d problem s a t i s f y the c o n d i t i o n s f o r r e l a t i v e e q u i l i b r i u m a t a s p e c i f i c i n s t a n t of time. When the same conditions are a p p l i e d to the four-body s i t u a t i o n then, corresponding to a sequence of time i n s t a n t s , there w i l l be a sequence of p o s i t i o n s of r e l a t i v e e q u i l i b r i u m .  The locus i n the £> n  plane  described by these points of e q u i l i b r i u m w i l l be c a l l e d a reference l o c u s .  +  T h i s material was presented at the 13th I n t e r n a t i o n a l Congress of Theoretical and Applied Mechanics, Moscow, August 21-26, 1972, and at the 23rd International A s t r o n a u t i c a l Congress, Vienna, October 8-15, 1972.  223 A2.2 I f the point s^, n  L  i s located on such a locus then, when the v e l o c i t y  and a c c e l e r a t i o n of p^ are both zero, we obtain  LEI  *  This device was mentioned, although not i n v e s t i g a t e d , by Steg and deVries [ 8  ] , and appears also i n the work of Tapley and Lewallen [ 9 In contrast to the r e s t r i c t e d problem the terms £ . and  ].  in  Equations A2.1 to A2.4 are functions of t , so the s o l u t i o n of Equations A2.3 and A2.4 w i l l also depend on t .  The method by which these  s o l u t i o n s can be found i s described i n the f o l l o w i n g s e c t i o n , but f o r the present we s h a l l assume that K^(t) and n^(t)  are known functions of t .  Now l e t 5  =  O-  (  +• x )  =  5^ +• roc  where a i s a normalizing constant, p  4  so that x and y define the motion of  with respect to the .reference l o c u s .  then be w r i t t e n  (A»-0  Equations A2.1 and A2.2 can  2214. A2.3  -  <v  a. +• \  -5,  7  4—  1  IT- |  3  -  <. = (  a  Z—i  i_  i  -  lit  (A  I f the term  LI  3  '  3  x.S)  i s expressed as  J  in-1  , .,-J r  and we expand Equation A2.9 u s i n g , f o r example, Chebyshev polynomials, then f o r Equations A2.7 and A2.8 we obtain  II)  where the functions r , A-j and  contain t e x p l i c i t l y and i n v o l v e terms  i k of the form x y J  (j,k  i s s u f f i c i e n t l y small  = 0,1,  2, ••••).  When the o r b i t a l parameter a  these equations may be w r i t t e n  225 A2.4  (A*.13)  where a , b and c contain t e x p l i c i t l y .  I f X Q and  denote s o l u t i o n s  of Equations A2.12 and A2.13 when the terms 0(a) are neglected,  the  approximate s o l u t i o n s of Equations A2.1 and A2.2 are given by  S = 1  *  %u  °-*o  +  (A2.I4) (R2..IS )  "lc  The character of the reference l o c i w i l l determine whether or not t h i s approach i s f e a s i b l e ,  so we should next attempt to solve Equations A2.3  and A 2 . 4 .  A2.2  Numerical Explorations At a p a r t i c u l a r i n s t a n t of time the p o s i t i o n s of r e l a t i v e  e q u i l i b r i u m are defined by the c o n d i t i o n s 3  3  ft  fe  Ir.;!'  These equations correspond to Equations 3.40 and 3.41 f o r the r e s t r i c t e d problem but, i n contrast to the equivalent r e s u l t f o r the  226 A2.5 r e s t r i c t e d case, i n general i t i s not p o s s i b l e to p r e d i c t i n advance the t o t a l number of reference l o c i .  D e f i n i t e r e s u l t s can, however, be  obtained i n one s p e c i f i c i n s t a n c e , and the approach i s described below.  A2.2.1  Intersections  with the E, a x i s at t = 0  When t = 0 the three primary bodies are c o l l i n e a r and l i e on the E, a x i s , as shown i n Figure A 2 - 1 .  Figure A2-1  Primary c o n f i g u r a t i o n at t = 0.  Note that t h i s i n i t i a l c o n f i g u r a t i o n i s chosen so that both p-j and p are located on the negative t h e r v f (0) = 0, and we have  - X  E, a x i s .  3  One s o l u t i o n of Equation A2.17 i s  227 A2.6 This equation can be w r i t t e n  (s,-O  f^-O"  x  <*,-0*  from which a seventh-order polynomial i n E,^ can be d e r i v e d . i e n t s of t h i s polynomial depend on the signum functions  The c o e f f i c -  sgn(£.j-£;^),  and i n f a c t there are four d i s t i n c t cases corresponding to the conditions  When Equation A2.19 i s expanded we obtain  —  228 A2.7  +  5  t  -1,  +  sC-c 6C7-) + ^ ^ s ( 5 , - O +• fx-S^^C^f  r  O = o  (fli.^o)  where  &(4> = - 2.  229 A2.8  We must apply numerical methods to obtain s o l u t i o n s of Equation A2.20, but t h i s w i l l require s p e c i f i c values f o r n , fi, C , u. and 5. (where i = 1, 2, 3).  As an approximation to the motion of p-j, p  p , suppose we take the model described i n Chapter 2 (Section 3  on  page  2.30  ),  with  the  a d d i t i o n a l r e s t r i c t i o n that p  i n c i r c u l a r o r b i t s about t h e i r centre of mass.  2  2  and  2.6, and p^ move  The l i m i t a t i o n s of  this  approach have been discussed i n the i n t r o d u c t i o n to Chapter 3, but any e x p l i c i t s o l u t i o n f o r the motion of the three primary bodies could i n general  be used.  An i d e a l i z e d model f o r the sun, earth and moon,  c o n s i s t e n t with these r e s t r i c t i o n s ,  can be derived from the f o l l o w i n g  expressions:  X = r Z3  n*  = X  3  L  S  r, = At X C  hx  =  r, - ( jry  R  3  H  230 A2.9  Symbol  Parameter  Value  Astronomical Unit  A  0.4990 04785 x TO l i g h t sec.  V e l o c i t y of Light  c  0.29979.25 x TO m. s e c . "  Ephemeris Seconds Per Tropical Year (1900)  s  0.3155 6925 9747 x 10  Earth-Moon Mass Ratio .  Moan  Fsrth-Mnnn  llictaiiro  9  m /m 2  3  3  0.813024 x 10  *  8  sec.  2  q  r  Z6  n 3«4-inn y i o m  PI Geocentric Constant  •: IT  Gravitational  3.1415 9265 3589 79324 Km  2  0.398603 x 1 0 Table A2-1  Astronomical Constants  1 5  m sec? 3  1  2  231  Parameter  C r  l  r  2  r  3  Value  0.3891 7127 26 X 10  3  0.3891 7245 58 X 10  3  0.1215 0313 96 X I D " 0.9878 4968 60 1.0  n  0.7470 2544 92 X l O " y  l  y  2  y  3  1  0.3289 2414 85 X 10  1  6  0.9878 4968 60 0.1215 0313 96 X l O "  Table A2-2  Normalized Constants  1  232  A2. where the q u a n t i t i e s A , c, s and Km^ are defined i n Table A 2 - 1 . numerical values f o r the astronomical  The  u n i t and earth-moon mass r a t i o  are  from Ash, Shapiro and Smith [57 ] ; the remaining constants are taken from Clemence [56 ] .  Corresponding numerical values f o r n , £2, C,  r.j and y . (i = 1, 2, 3) are given i n Table A2-2. Now that s p e c i f i c values f o r the parameters of Equation A2.20 have been obtained, we must s e l e c t a numerical method f o r s o l v i n g n ' t h order polynomials with real c o e f f i c i e n t s . difference  algorithm [ 6 5 ]  Rutishauser s  quotient-  1  provides a p a r t i c u l a r l y elegant method of  s o l u t i o n , and has the d i s t i n c t v i r t u e that convergent approximations all  to  roots are derived simultaneously. There a r e , i n f a c t , four d i f f e r e n t polynomials to s o l v e ,  corresponding to the cases defined on page A 2 . 6 .  Roots o f Equation  A2.20 f o r these four separate cases were found using a version of quotient-difference  the  a l g o r i t h m , and the s o l u t i o n accuracy was tested by  evaluating the polynomial with these s p e c i f i e d numerical v a l u e s .  The  r e s u l t s are given below.  Case 1  £  L  > 0.12150 31396 x 10'  •1  REAL  COMPLEX  O. 39 O 2.  S"8 3 4  -*.  io  0.39 02.  S834  x  io"  O .1363  66 3 2  O. 7 - 9 4 9  l9 9o  * io"  0 . 1 8 6 3 6632.  O. 7 - 8 4 9  19 9 0  *  O . 490 0 2 8 ? €  0 . 8S57  Il o I  O .4-9 o o  O . 8557  If o I  O . IOO  3  287-8  l 39 I 3  x  Io'  0.2.905  I 841  - 0.7-<\OS  O.o  x  18 4 2  i o  1  K IO  Z  1  io"'  233 A2.12 Case 2:  -0.98784 96860 < £. < 0.12150 31396 x 10" REAL  COMPLEX  - o. 3102.  S"834-  - o. 3°02-  S"8 3 4  ~ o. 1  D 62. 5 ? o 3  - o. |  0 6 2  -o. - o.  X  io3  O -  1  1 062. 5"?o3  x  S?oS  - 0  10 ' .o  .  0 .  1  — 0  .0'  0  8372,  -0.38971 24558 x 10° < K < L  < to"  - o. 3 l O i 5834- * vo -  o . lozz T-i 1 I  -  o . <\07-Z. O.  5o3S  A-5SI  O.  Ilb3  <?4-8o  ?  L  31 3 2  I4S3  3l 3 8  I ft S 3  31 3 8  I4-S3  31 3 g  1  . O  -0.98784 96860  3  -o.ZloS"  - o  -  7  184-2.  O . I 3>06  * 10  K 10 "  0. X°ioS  5  O . S03S 4-581  Case 4 :  (o -  COMPLEX  o . 3<jo2  -  I 4 ? 3  .  REAL -  l S 4 2  s  - 0 • Z1 o5  10 <  - 0 . 8360  Case 3:  O . 2-10  57o3  1 0 6 2  1  . 1306  x IO*"  7<?78 71 7 8  o .  ? 6  4-8  o . 8 6 04-  76  4- 8  O - O  1  < -6.38971 24558 x 10  3  COMPLEX  REAL - O . 35€«  o 2 1 I * 10  - o. 4 l ? l  Zo3fc  x io  - o. 4-44-4- 73S"S  x 10  0.0  3  o.o  3  0.0  1  O. 3627  14-74 x io~'  o . 36. 8?  I 4 74  - O . <?8 6<?  oo$l  - o. 1861  oo$i  x io"'  -  -  O . 472k  3©4I  o . 4-72-6  3o4-l  O. 5 Z 8 3  o 16 2  K  O. S Z  O l6 Z  xi 1 o  8  3  10'  23k A2.13 In a l l cases the residual e r r o r was below 10  -15  i n magnitude.  Note that  there i s no ambiguity i n these r e s u l t s , as the complex part of each root i s e i t h e r zero or a w e l l - d e f i n e d quantity s i g n i f i c a n t l y d i f f e r e n t from zero.  Now only those real roots w i t h i n the regions of v a l i d i t y f o r £|_  a c t u a l l y define points on reference l o c i , so t h a t we can s t a t e the f o l l o w ing  result:When t = 0, f o r the p a r t i c u l a r numerical model chosen here p r e c i s e l y four reference l o c i i n t e r s e c t the £ axis.  The points of i n t e r s e c t i o n are  €  =  0.1001 3913 x 10  K  = -0.8360 8372  K = -0.1153 9480 x 10 €  1  1  = -0.4173 2036 x 1 0 . 3  I t might be a n t i c i p a t e d that three of these i n t e r s e c t i o n s be located near c o l l i n e a r Lagrange points of the corresponding problem.  From Szebehely [ l  1 8  ] , when  points are located (approximately) £  = -1.15568  K = C  =  = 0.01215  restricted  the c o l l i n e a r  at  (1^)  -0.83692  (L )  1.00506  (L )  2  3  ,  so that the conjecture i s j u s t i f i e d .  This r e s u l t c a n , however, only  be a p p l i e d when Equations A2.16 and A 2 . 1 7 reduce to the form of Equation A 2 . 1 8 , which i s true only when (n - ft)t = 2kir, where k = 0, ± 1 , ± 2 , •••• .  will  235 A2.14 A2.2.2  P o s i t i o n s of e q u i l i b r i u m when t f 0 The functions f ^ ( t )  and f ^ U ) i n Equations A2.16 and A2.17  define two nonlinear a l g e b r a i c expressions, but the formulation i s not particularly helpful.  then F ( £ , n , t ) > 0  If,  however, we take  and a s o l u t i o n f o r £ and n which s a t i s f i e s  the  condition  at a s p e c i f i e d instant of time w i l l a l s o s a t i s f y Equations A2.16 and A2.17 at t h i s time i n s t a n t .  Fletcher [ 6 6 ]  has developed an e f f i c i e n t  algorithm to determine the l o c a l minimum of a nonlinear algebraic f u n c t i o n of n v a r i a b l e s i n the v i c i n i t y of an i n i t i a l estimate, which can be a p p l i e d to solve Equation A2.22 f o r s p e c i f i c values of t .  The  extension of t h i s procedure to generate a complete locus i s summarized below. 1.  At t = 0 s e l e c t an i n i t i a l estimate of the p o i n t on a particular locus.  2.  From t h i s estimate, determine the exact l o c a t i o n of the point using F l e t c h e r ' s method to solve Equation A2.22.  3.  Increment t by A t .  4.  With the preceding point as an i n i t i a l estimate, the process by r e t u r n i n g to step 2.  iterate  236 A2.15 In t h i s approach we must, however, assume that some point on the p a r t i c u l a r reference locus s a t i s f i e s t.  Equation A2.22 f o r every value of  There i s some evidence to suggest that F ( £ , n , t ) = 0 cannot be  s a t i s f i e d f o r a l l values of t along two l o c i , but a d i s c u s s i o n of t h i s d i f f i c u l t y w i l l be postponed u n t i l Section A 2 . 3 . Six l o c i were found f o r the sun-earth-moon system by t h i s method, using the approximate primary model of the preceding s u b - s e c t i o n .  For  each i n d i v i d u a l point the f u n c t i o n F ( £ , n , t ) was scaled so that F = 1 with the i n i t i a l estimate of the minimum.  F l e t c h e r ' s algorithm was -13  terminated when the p o s i t i o n c o r r e c t i o n was l e s s than 10 -20 which i n a l l cases reduced F below 10  .  i n magnitude,  Four of these l o c i emanate  from the points of i n t e r s e c t i o n given on page A2.13,and the remaining two l o c i correspond to e q u i l a t e r a l t r i a n g l e points of the problem.  The locus containing the point  forms a c i r c l e i n the £ , n.  restricted  £ = -0.4173 2036 x 1 0 ° , n = 0  plane so that the point of r e l a t i v e e q u i l i b r i u m  l i e s on a l i n e j o i n i n g the sun and the earth-moon mass centre. Results obtained f o r the f i v e l o c i near the earth-moon system are shown i n Figures A2-2 to A 2 . 6 . 1_2 are s i m i l a r i n character, and Lg.  Note that the two l o c i near L-| and  and so a l s o are those associated  with  The locus near L ^ , shown i n Figure A 2 - 5 , i s b a s i c a l l y  equivalent to that found by Tapley and Lewallen [ 9  ] (Figure 9 i n [ 9  although t h e i r primary model includes the i n c l i n a t i o n of the earth-moon o r b i t a l plane at an angle of 5.15° to the e c l i p t i c and takes i n t o cons i d e r a t i o n the e f f e c t of s o l a r r a d i a t i o n pressure.  The u n i t of time  i n Figures A2-2 to A2-6 i s chosen so that new moon corresponds to the instants t = n , where n = 0, ± 1 , ± 2 , •••• .  As one might a n t i c i p a t e  ]),  237 A2.16  1.  l  1  -\-\SS  -1-154  h  -0-2  0-4-  Figure A 2 - 2 A2-2-1  Locus near L^ Polar  representation  238  I  A2.17 -1-154-  -r  •1 -tSS  1-156  1  t  OS Figure A2-2-2  0-?  E, component  -1  o -is -4  1 o-S  o-s  -J  Figure A2-2-3  n  Figure A2-2  component Locus near L^  1 -o  t.  239 A2.18  r  0 . 2  V  h  -0.837  o.i  -0.836  -0.1  *• - 0 . 2  Figure A2-3 A2-3-1  Locus near L, Polar  representation  1  0  22+0 A2.19  5  -0.836  i  -0.837  H  L  -0.838  r — 0  0.25  Figure A2-3-2  £  > 0.5  • 0.75  .  t  1.0  component  0.2  0.1 10'  0.25  0.5  -0.1  -0.2  J  Figure A2-3-3  n  Figure A2-3  component Locus near L  0.75  t 1.0  214-1 A2.20  1.0  0.75  H  0.5  0.25  -i 0.5  0.75  -0.25  -0.5  -0.75  -1.0  Figure A2-4 A2-4-1  Locus near L^ Polar  representation  1.0  1 1.25  2i+2 A2.21 1.0  0.8  0.6  0.4 0.25  Figure A2-4-2 1.0  '  -T-  £  0.5  0.75  t  1.0  component  1  0.5  ^ 0.25  -0.5  -1.0  0.5"  0.75  H  J  Figure A2-4-<3  n  Figure A2-4  component Locus near L  3  1.0  t  0.8  0.7  0.7  •0.5  -0.3  Figure A2-5 A2-5-1  -0.1  Locus near L^ Polar  representation  0.1  A2.23 0.1  1  0.1  i  0.3  1  0.5  1  0.7  1  1  0  0.25  Figure A2-5-2  1.0  1 0.5  1 0.75  1  t  1.0  E, component  n  0.9  0.8  0.7  -\ 0.25  Figure A2-5-3  n.  Figure A2-5  0.5 component Locus near L^  0.75  1.0  t  1L  •0-7  -o-3  o-S  o-i  -o-l  -0-7  H ro  -O-S  l-O  Figure A2-6 A2-6-1  Locus near L^ Polar  representation  ro -p.  2i;6 A2.25  -  0'\  0-3  i  -OS  - 0... 7  —i .0-2.5  Figure A2-6-2  £  1  r-  1  0-5  0-75  i-o  t  component  -0.7  -0-8  -o.<\  J  - i-o  H  t  i  o-S  O'XS  Figure A2-6-3  n  Figure A2-6  component Locus near L,  •o  2k7  A2.26 f o r coplanar primary o r b i t s , a l l f i v e l o c i near the earth-moon system are p e r i o d i c functions of time.  A2.3  Limitations One p o s i t i v e r e s u l t  of t h i s a n a l y s i s i s the extension of Tapley  and Lewallen's i n v e s t i g a t i o n [ 9 ] to include a l l p o s i t i o n s of r e l a t i v e e q u i l i b r i u m f o r the sun-earth-moon system at a p a r t i c u l a r time A number of f e a t u r e s ,  instant.  however, render the reference locus approach  u n a t t r a c t i v e i n the a n a l y s i s of four-body motion. (i)  The reference l o c i only apply to one s p e c i f i c primary model, and i f any parameters are a l t e r e d the  entire  a n a l y s i s must be repeated, (ii)  When the o r b i t a l parameter a = 0, the functions  £^(t)  and r i ^ ( t ) are not s o l u t i o n s of the equations of motion (because the q u a n t i t i e s E,* and and A2.8 are 0 ( a ) ) . - 1  i n Equations A2.7  Although a may be s m a l l , the  l i m i t process a -> 0 cannot, t h e r e f o r e , j u s t i f y neglecting terms 0(a)  be used to  i n Equations A2.12 and  A2.13. (iii)  An expansion of the term  _j  (where i = 1, 2, 3)  leads to a form of Equations A2.12 and A2.13 which cannot e a s i l y be s o l v e d , p a r t i c u l a r l y i f s o l u t i o n s are to be accurate over large i n t e r v a l s of time, (iv)  Not only the l o c u s , but also i t s f i r s t and second d e r i v a t i v e s , must be evaluated.  The  process  of  differentiation  214-8  A2. i s i t s e l f s t r a i g h t f o r w a r d , although the computation time increases,  but  rapid  changes i n C[_(t) and Hj_(t) present  severe obstacles to any accurate a n a l y s i s . i s evident in the two l o c i near and A 2 - 6 ) .  This behaviour  and Lg (Figures A2-5  An attempt to continue the l o c i through these  abrupt t r a n s i t i o n s d i d not succeed because no s o l u t i o n f o r Equation A2.22 could be found.  The only inference which  can s t r i c t l y be drawn from t h i s r e s u l t i s that algorithm f a i l e d to f i n d a l o c a l minimum, but i t  Fletcher's appears  l i k e l y that no s o l u t i o n of Equation A2.22 e x i s t s in these regions of the  n  plane f o r c e r t a i n small i n t e r v a l s of  time. T+  i.fac  Aar-A  A r\A  K n r a n r n  r\-f  + h "i c  arr-nmul ati'rin  rt-F r l i f f i n i l  f i  abandon the a n a l y s i s i n v o l v i n g reference l o c i and concentrate instead on a l o g i c a l development from fundamental p r i n c i p l e s .  2k9 A3.1  • APPENDIX III — SOLUTION OF THE THREE-BODY PROBLEM  With the exception of three d e r i v a t i o n s (which are given below), the a n a l y s i s of Chapter 4 up to Equation 4.83 requires l i t t l e f u r t h e r amplification.  We now continue the development beyond t h i s point to  / -14\ obtain s o l u t i o n s uniformly v a l i d f o r t = 0(e ). A3.1  The Homogeneous Solution I f we take x-j = e, x  2  = e, x  3  = f , x^ = f , then Equations  4.11  and 4.12 can be w r i t t e n in the form  o  I  3  o o  o o  -X  o o o o  o X  (A3.f)  I  o  which i s equivalent to ^  =  A  where x i s the vector (x-|, x , x , x^) . 2  transform of x ( t ) , 5E.Cs:>  3  I f x(s) denotes the Laplace  then from Equation A3.2 we obtain = ( s i - A ) -i 51 C o )  .  CM. 3-")  Now det(sI-A) = s (s+1), and a f t e r some s t r a i g h t f o r w a r d manipulation:  250  A3.2  4  01-/0"  - 2±_ S--r\  z  s  **•*•• -  6  6  2s  - 2L  s+r l  *Vl  S  "  3- _  J_ .  A-  -  3  4-S  -  3_  S  - 2.  6$  o  o  s  0*3.4) Taking the inverse transform of Equations A3.3 and A 3 . 4 , we can w r i t e  xCt)  = <f>tl) x C o )  ,  where 0 . - . 2.<<ofc  4 - 3<<r>t  3 *.vfc  Cos b  (A3.S) X  4*J^b - 3 1  Equations 4.13 and 4.14 now f o l l o w d i r e c t l y from Equation A 3 . 5 .  A3.2  The Non-homogeneous S o l u t i o n For the non-homogeneous case we have  ii - Z v ' - 3UL  =  £  (t)  (A3.0  2^1 A3.3 Cfi-5.7)  where the general form of E* and F* i s given i n Equations 4.41 and 4.42. If B denotes the vector (0, E * , 0, F * ) , Equations A3.6 and A3.7 can be T  w r i t t e n i n the form 5. =  and  P5c •+ B  therefore s£s)  = _(sJ-A)" x('o) V  +  (si -A)'  1  where B(s) i s the Laplace transform of B ( t ) .  BC*s">  When B(s) i s known, the non-  homogeneous part of the s o l u t i o n corresponding to the term (sI-A)~^  B(s)  can be derived from Equation A3.4. I t i s convenient to p a r t i t i o n Equations 4.41 and 4.42 so that periodic terms i n v o l v i n g wt are considered separately.  If e ^ ( t )  and  f^l^(t) denote the non-homogeneous s o l u t i o n s of Equations A3.6 and A 3 . 7 , then f o r  I F * Cfc)  +• b,b . t . b ^ t . + >j«ot.  we obtain  4- ( Z b , - 2 b ^ - a ) ^*b 0  3b s;».b  (fl  3.8)  252 A3.4  -  [  +  _ L  3 J . J l> *  -  fe,  J  t  3  x  + ( ft + 2 - 0 b *~b  +  t  (  Cft^O  - * - 0 t *=,b  S i m i l a r l y , when  the corresponding non-homogeneous s o l u t i o n s a r e :  [ X3L + o t {  0  b  -  I su.«jb  3 f  to  %3-  t  -  (3+w  1  5''  cost (A3, io")  x  0  b  (B3.il)  ZkJC  ItOC.  , cJ b  Q  Co -0  L  -  :, - %u3d~  +  0  +  onb  41.  ifa>C  0  + (.3> + «J  cos <~>h  253 A3.5 A3.3  Expansions of sin(t+8) and cos(t+8) From Equation 4.32 the expansion f o r 8 i s of the form  so that we can write ««. ( b + P )  =  suvb or>9 + c«fc  <*> CsQ, + e" e ^ . - - ) + c Q ^ C e P , v t ^ P ^ ...  + <i,fcJVs^P  )J  1  <>  tr>  t>  and, s i m i l a r l y ,  - COS b  - 5.K  o o C f e P , ^ ^ P , . * ••• > - s : ~ P s i ^ C t P , 0  ••• *)  j  b £ * ~ P co C eD, * G" 0 _+. • ) -t- Cr,t904^C 69, + t i e v + - •) ^ 1  0  i  0*3.13")  2 2 The functions sinCeS-j+e e * * * ) and cos(e8.|+e 8 «««) can be expanded as 2  +  2  •51  .(AVI4)  1  -  e^f* z!  CA3-15-)  25k A3.6 We therefore obtain f o r sine and cos9:  8  -  I  P  x  ^  9  X  0  o  - e," ,^P 1  0  + e  (  x  C  qo 0  0  X  and Equations A3.12 and A3.13 can now be evaluated without d i f f i c u l t y .  A3.4  Continuation of the Uniformly V a l i d S o l u t i o n The sequence of p a r t i a l d i f f e r e n t i a l equations defined i n Chapter 4  (see Equations 4.35 to 4.40)  can be continued as f o l l o w s : •  -7-2^  -  3a,,  =  - z  it  1  (fl3.l«)  255 A3.7 = if-  - X  '5  i\ it?-  it  OCe  1  I  it  - Z^,„ - 3 a  l e  =  - 2v  - Z  it  [ 1 it""-  J  >)  -  OCfc ") 1  "f  2^  5  7  J OCt  OCe*)  xo-j-o .  oCc  + z ^ Vs  -  s  itZt;  L  (A3-19)  5  ^.5  +  OCt'")  1 if*-  °t  itz-c  i ri oo)  (A3, io) Ofl)  - 2  d*V$  J  -+• 0)0.5  J <?Cfe*)  "(3.XV-)  +  p.  <'-/**) J O C t ' ° )  oCt)  (fl*.ai>  256 A3.8  -  it  •  i — —  + 5" }\*? X iC  + bj* ~l s  i t  J ^ J ^ )  +5 1  -  7  ^  at 1  -  Joco  _  it* •  - 2  +-  .  L it*  at  —  b <j  s  itir  ^ J o C O  I Jb*  5V  Joco J  OCi)  257 A3.9  it* it  Z G - ^ O  L  J  -  z  oCe'*\)  —  i^J-f  OCi ) 6  1 Xt*" Xt ) -  I  o C 6  ^>  1  bb  it*  bt • d^Socrf  it  L  it Sou)  0-»o  Li ^,5  dfc»-  -2-^  ( >  it  o  t  f  e  3 )  lit*- it J  oC.)  J  —  . «3b^r I V  L*t*  J  J ° ^ " 0  z  '.•I.  bV  Ttir  atjoc£<0  J  (_ i t ^  i b  OCeO  atar  oft . 1  CA325)  jfc  - 3u  258 A3.10  _I  i  it"  it joceO  at  ^j^)  L. * ^  ^  dt  L ^  ^  J o^)  it  JoCe^  L at*  at J oco  ©CO  £y  -  +  l3  at*  F  it  -  I  J>  LTb*  2.  it )t J  • 5 r  v  o  f  i  t  0  o  i at*  a O  at  La t  1  ^a-i? a r  .at J a ) &  £.  T  L <>t  ht  v  ; ' l ^ J o ( 6 ' ) •' The q u a n t i t i e s Al.8 E  g  L a t - ••  aba*} o o o  «fc J  t- " dtl  and F. are defined i n Appendix I ,  oCO J ..^Joco Equations  and A l . 9 , and f o r . E g and Fg i n Equations A3.18 and A3.19 we have  = Fg = 0 .  and A3.19.  Now consider the remaining q u a n t i t i e s i n Equations A3.18  A f t e r d i f f e r e n t i a t i n g Equations 4.25 and 4.26 we o b t a i n :  259 0>\t  - X  A3.11  —  s  ibiv bx  bx  2t S*-  - Z  (b+£>3/>_  . + ^>JLg  5  obtr  bv + a-cr  2t?  bv  which can be rearranged i n the form:  2  -.1  - 2v  =  s  at  2 s.«.  2-5^  z  b  v/ •+. s  ^a •  t  Zcos t  -  61  bfi_  5  ^  f  cose  ^5  Si^9  -2-cost ^ (b COS 8 - ph9 SLVLB  <r  1  b"C  At the present point i n the a n a l y s i s we have shown that | | = 0 ( e ) , f j = 0(e ) and | f = 0 ( e ) . 3x 9T 3  ~bt  +6*  3  59^  i 3/>, -f> ^bK 0  I  ar  With these s i m p l i f i c a t i o n s :  2  QoP - p>z® bV  -/j.e.af*  *«. 6>.  -/s p,5p 0  ^  = 0(e),  3  < C«0 Po  2.  b  260 A3.12 3& c*>9 t  D  + e•  +6*  l  }v  ytf  zx  2.  3  Equations A3.28 and A3.29 can now be evaluated, and hence  2 5^,  - 2-  + 2 cob"  -  6t  ^  [I  CA3.32)  to  CA3-33)  + 2s;*.fc  [ t  5" <?/s" -/S.P,5fix  0  o  +  ]  261 A3.13 From Equations 4.25 and 4.26: -  5^t  4-  5 •Zy~!r  + Ze^fc  8  -  -  (G3-34-)  e  2-^  l  1  Six-  do  S'".  i  (.AS- 3 5 )  I  I  3 t £fk  and we tnen obtain  A  F  ]  /*. F I t has already been shown that  ||ii  s  constant  quently both these q u a n t i t i e s are zero.  (on page 4.20) and conse-  When Equations 4.25 and 4.26  are d i f f e r e n t i a t e d with respect to t : — 3 v"  (A3.36)  5  -  bt  ^ v - C t +p)  (ft3.3?)  262 3.14 and from these expressions:  2fu. - Pv $  s  (A-3. 3S)  ib> it  oCt)  (A3.  The terms  1 2 a V -4- }u. ~) L at-av atav AAt? V J .  and I  s  a* J  at it?  s  are both  31)  0(e),  and therefore do not c o n t r i b u t e to the non-homogeneous s o l u t i o n of Equations A3.18 and A3.19.  For Ug(t) and v ( t ) we have (from Equations g  4.60 and 4 . 6 1 ) :  «?V  6  - Xr  it*  it  if -  it  1  iVo Zb  b  2c<o b - _£  CA3-40)  (A3-4-0  and the non-homogeneous terms can now be accumulated.  The r e s u l t i n g  expressions are:  _*  =  %H (\+H>VC\-^')\ y  £ c»2.b -£  \  +  6(i +/ 0 \ « * ,  +  zip,  - 6b£^ i-v  I  ar  L  ai?  J  iv  3  263 A3.15  + Ceo t J  I  air  + *0  -  ?^,Ci+fOVi-jO  iX!  +  5;~b  f 2PA,  '• *-  dX  Jx  Ceo b  CA3.43)  When a-| s a t i s f i e s Equation  4.58:  so that Equation A3.42 s i m p l i f i e s s l i g h t l y .  The symmetry of terms i n E*  and F* i n c i d e n t a l l y provides a useful t e s t f o r e r r o r i n these f u n c t i o n s .  26*4. A3.16 I f terms i n v o l v i n g t  2  are to be eliminated from the  non-homogen-  eous s o l u t i o n , from Equation A 3 . 9 :  o  which j u s t i f i e s the assumption that We have already shown that s i n 0  Q  i s constant (see Equation 4 . 8 2 ) .  = 0, and with t h i s condition on 6  containing t s i n t and t c o s t are eliminated from u  g  and v  g  Q  terms  if  iv  0+fO'>> -  «»9o\ L  From Equation  p W* 0  iV  d~C  (A3-4-0  4.76: (A  3-4?)  it  and therefore Equation A3.45 takes the form  16  Xv  (A3-4S)  The only value of 9^ f o r which 3g does not v i o l a t e the uniform v a l i d i t y condition i s 9, -  consequently o  CP 3 . 4 0  i n which case  air  (A3 s-o) r  265 A3.17 Equation A3.46 can be rearranged as  and i f we again use Equation 4.76:  •  since cosBg  f  0.  The q u a n t i t i e s  and  BQ»8-J  6Q  are a l l constant, and  therefore  <^3  =  constant  .  At present we have no way of d e f i n i n g  3  N  > 3  N  ,  these q u a n t i t i e s are known to be constant. that both 9  2  and 6  3  and  , although  I f , however, i t i s assumed  are constant, then from Equations 4.81 and A3.49  we obtain  where s i n 6  Q  the q u a n t i t y  = 0.  The one remaining secular term i n v ( t ) must e l i m i n a t e g  -e (3ta ) 4  4  i f v ( t ) i s to be uniformly v a l i d to 0 ( e t ) . 5  Consequently, from Equations A3.9 and A3.11:  5  266 A3.18  and, i f we s u b s t i t u t e f o r the quantity  t o s 9 1 / ^ f ^ + /*, ^ P r i from Equation  A3.46, t h i s can be w r i t t e n as  From the argument of page  4.15:  I t i s , i n f a c t , l e g i t i m a t e to assume that  and 6 ^ are constant  (see  Equations A3.83 and A3.170), so t h a t , from Equations A3.53 and A3.54:  assuming  is  constant-  E x p l i c i t expressions f o r Ug(t) and v ( t ) g  are not needed at t h i s  stage of the a n a l y s i s , and we can postpone evaluating these functions u n t i l the s o l u t i o n s f o r u-j and v-j are considered. 2  2  Now we proceed with the s o l u t i o n of Equations A3.20 and A3.21 for u  1 Q  and v  1 Q  .  This p o r t i o n of the a n a l y s i s i s p a r t i c u l a r l y i n t e r e s t i n g  because here nonlinear terms f i r s t enter the sequence of p a r t i a l d i f f e r e n t i a l equations.  From Appendix I , Equations A l . 8 a n d ' A I . 9 :  267 A3.19  ]  (.A3. SrO  (03.58)  L  g  |6  and from Equations A3.28 to A3.31 we have, with 0^ = s i n 0  = 0:  Q  COS I  + 2-eosk  -  C<JD 9,  ^/>x^2- 1  >f A > ^  a  ar  £ /j as<, +/\a^2 1 av ' PT- ' j . 1  - 2 cost  General expressions f o r and A3.35. and  4  or  2 2o  o*<i and d V * are given i n Equations A3.34 s  A t present, however,  = o(e ),  (ft  0  a t; _  so that  z  ]>jf$ at*  ||=0(e ), 4  and }j*s bx  |f=0(e ), are both  x  i t follows that:  * 0Ct»)  oU ) 1  0(e )  2  o  4  0(e ). 4  Consequently  268 A3.20 From Equations 4.47 and 4.48:  -t- o a )  (ft3.6l)  6  (A3.62)  since 0-j = 0 and a  Q  = 0 (see Equation 4 . 5 2 ) .  The nonlinear terms i n  Equations A3.20 and A3.21 can now be evaluated, and we obtain  • ( 3 / - - feu.*) a  -  9 //** o cos 2.b  ( A 3.63.)  I (l- /*>)  (see a l s o pages A3.26  to  A3.39 f o r a d e t a i l e d d e r i v a t i o n ) .  When sin0 = 0 Q  and 0.| = 0, the equations f o r sin0 and cose s i m p l i f y t o :  OcV)  «~ 9  Cos »  L  2.  .  0  (A3.65)  269 A3.21 (see Equations A3.16 and A3.17 on page A3.6 ), so that s i n ( t + 9 ) and c o s ( t + 9 ) can be w r i t t e n :  s.v. Ct + B)  5 ~ t | I - £*fC ^  + cosh |  ocv; CO ft, (A3.iS)  We then o b t a i n , from Equations A3.36 and A3.37:  r ^b*  at  — (i e^,9 east e  0  +- £ |^ 3<*, -  C«0  t  - *~b  ~* <<o b J  270 A3.22  3b*  L  at  4- 0 ( t  f  )  ,  and therefore  cIVj  - a>/  ^  3oi  s  -  x  &o  at  - <*>9,  033-92)  From Equations 4.25 and 4.26: z. dldX  -  7- S~^s  +  a t a r  a_u.  $  aj/  =  5  -at a»t  atr  a-r  a_t<  %  5  f/  3  (A3.7S)  ar  ^ t + O  - 3 / i s i ^ C t + a") ' a-r  ar  - A- < u <ar  2 ^ ;  dx~  L  a-u" -  3x  1  - 3ta ar*  L  ar  ~ixr  2to$Cb-*-e) a-cj  2s^b+e)  (A3.?6)  271 A3.23 At the present stage of the a n a l y s i s , i n a d d i t i o n to the order r e l a t i o n ship given on page A3.19, we have and  ^lif = 0(e ) . dr -  ||> = 0(e ),  = 0(e ),  4  4  9x  0(e ) 4  None of the expressions defined i n Equations A3.74  2  to A3.76 therefore makes any c o n t r i b u t i o n to Equation A3.20 and A3.21. From Equation 4.69  u ( t ) = Vy(t) = 0, and a l s o 7  (see Equations A3.40 and A 3 . 4 1 ) , so that we can now accumulate the nonhomogeneous terms i n Equations A3.20 and A3.21.  The corresponding  f u n c t i o n s , E* and F * , are given below. 6b  L ' + o o b" C«>9/  + . QoZb  :*  -  ar  L * r  atr  '  a  t  d - ^ O J  J  '  1 f*,(>-fO 4-  L iX  It,  2. 9^-^  IT  1^  i  TTT^ ( A3.?« )  - 35 Y C<X  S£~4-b  272 A3.24 To eliminate secular terms i n v o l v i n g t for v  from the non-homogeneous s o l u t i o n  :  1 0  -  which j u s t i f i e s  °  (A3.rO  the assumption i n Equation A3.56 that  is  Terms containing t s i n t and t c o s t are eliminated from u ^ ( t )  constant. and  v^(t)  if:  -  f>„)K  f>o^->-\ L ar  +  ar  C^—  y  +  br  J  fc\  ^  I >r  '  (AS.so)  =• °  ?  J  We can again s u b s t i t u t e f o r the quantity  - °  (Ai.sO  •  iP> - ( l + ^ ^ ^ J  from  Equation A 3 . 4 7 , and Equation A 3 . 8 0 then takes the f o l l o w i n g form:  The exact values of 3 Q and 8  2  have been i n doubt since Equation 4 . 8 0 .  Now, however, t h i s uncertainty can be resolved because, unless Q  z  = O  (A3.S3)  the f u n c t i o n 8^(x) v i o l a t e s the c o n d i t i o n f o r uniform v a l i d i t y i n x. When e  2  = 0 we have  273 A3.25 i/ ^3  =  O  •  0*3.84)  Equation A3.81 can, i n t h i s case, be w r i t t e n :  and, since $Q,3-|,$2 and  H.*  =  £f>3 are a l l  constant  constant:  .  (fB.sO  I f we now assume that 8 and 6 ^ are constant;, 3  then from Equation A3.85:  (see Equations A3.170 and A3.211). The remaining secular term i n v^g(t) i s generated by the expressions  i n Equation A3.77 and  16  i n Equation A3.78.  I f we assume t h a t a_ i s constant  (see Equation  b  A3.167), t h i s secular term must e l i m i n a t e the quantity - e ( 3 t a g ) i n 5  27k A3.26 the expansion of V c ( t ) .  By the usual argument  O  From Equations A3.9 and A3.11 we then have:  (page 4 . 1 5 ) :  ,  (A3.S8)  and, a f t e r s u b s t i t u t i n g f o r B from Equation 4 . 8 1 , Equation A3.87 takes Q  the form  31  + 2ZS" S12-C1 +  Note that Equation 4.81 f o r 3  Q  0  i s now known to be v a l i d (since  = 0).  and consequently the s p e c i f i e d value o f 3 can l e g i t i m a t e l y be used Q  i n the a n a l y s i s . Before proceeding with the s o l u t i o n o f Equations A3.22 and A3.23 f o r u ^ and v ^ , i t w i l l be h e l p f u l to evaluate the nonlinear functions  and  e' C3u.v^ 2_  in detail.  2-Cl-fO  (which i s rather tedious) i s only continued to 0 ( e ) . 1 4  This  process  275 A3.27 We have, from Equations 4.27 and 4.28:  -  e*^  +  fe'^  4- e*v  f  + 6*v  ....  8  .  («»,iO  Some e f f o r t can be avoided i f the c a n c e l l a t i o n of secular terms i n v.-(t) o  i s taken i n t o c o n s i d e r a t i o n .  In t h i s case the quantity - e ^ t o u )  is  omitted from the expansion f o r V g ( t ) , and corresponding secular terms i n v^ .j(t) +  are ignored.  The p r a c t i c a l consequence of t h i s s i m p l i f i c a t i o n  i s t h a t , instead of taking  v  5  =  2 ^ 5 ^  + ^  +  y  -  itcL,  when expanding the nonlinear terms, we use  and modify the terms Vg, v ^ , Vg, e t c .  to exclude the secular c o n t r i b u t i o n .  Equation A3.92 can only s t r i c t l y be j u s t i f i e d up to a c e r t a i n order, and to be precise we should w r i t e , f o r example:  L «  J  1  '2.  3*  J  276 A3.28 From Equation 4.69  u (t) = v (t) 7  7  = 0, so that we have:  Each term in these equations must now be evaluated. Equation  For u ^ ( t ) , from  4.25:  and we obtain  Now, from Equations 4.52 and 4.75:  and therefore  The expansion f o r 6 does not s i m p l i f y (because Equation A3.86 cannot, at present, be j u s t i f i e d ) , and so  277 A3.29  A l s o , from Equations A3.98 and A3.100:  d.f>  =  £<*xf>o *  +  6  * [  ,4  i/J,u '' 3/*o] +  +  8T  0  .( *^ &  •  CA3.ID2)  When G = 0, Equations A3.67 and A3.68 f o r s i n ( t + 9 ) and c o s ( t + 6 ) reduce 2  to  i^Ct-h9y =  cas9 si^b  cos  cos & cost - e  0  0  9^  +  z  6?  CPS0 c o s t  +  O  3  to$9  0  s^fc  O f f c O  tXfc*-")  (/u.ioj)  ,  From Equation A3.104 we therefore have  which may be w r i t t e n  c^Cfc + e)  >  _L  ( i+• coZb")  - e ^ s ^ z b  -t-  oft } 4  (AS.IOO  2 since cos 6 = 1 (see Equation 4 . 7 8 ) . Q  We are now i n a p o s i t i o n to expand  Equation A3.97, and the r e s u l t i n g expression i s given below.  278 A3.30  + fc" ^ 4- et, " + 1  +  1  + Oft*)  / 3 ^ ^ ( l f o o Z t )  -  4- * f i t n 9 t  t  0  cosh  .  J  CflrioO  From Equation 4.60 f o r u (.t) we have: g  2  and  /*|Cl-^i  - /3<to(b + e) ^ j" 2 Cot - Cco^b - 1 j  ,  (ft 3.107)  using Equations A3.98, A3.100 and A3.104 t h i s can be w r i t t e n  2 n.j l i j  =  2 . ^ , 0 - / 0 ^  / 3 «r>9 £ - 1 + 1 c o b - coj2fc + ± cos 3 b 0  5  0  £ 2 * , | 2oob - Qo2tr - 1 ^ + / S , & o £ ^_ 0  "  ^  ]  + &  The remaining terms i n Equation A3.94 can r e a d i l y be evaluated, and from Equations 4.60, 4 . 8 3 , A3.100 and A3.104 we o b t a i n :  279 A3.31 ltvr>4t  (fl-3.|0<j)  Crfo^t +• Z £ c o 4-b 64-  (A3.\io)  u..  2-  1^  87 64  -  7 coot +7. " 5 - 4 -  With the s i m p l i f i e d expression f o r v  ^ s ^ t + p )  + 4-^^s-^t+e; +  ^  c  (see Equation A3.92)  ,  and from Equation A3.104:  because cos 8 ^ = 1 .  The expansion f o r y i s of the form  (A*.!! *) -  so t h a t , from Equations A3.100 and A3.103, we can w r i t e : +9)  +  ^  ^ [ ^>^  =  +  ^(Zo^o  /V>  +  cos 9  / V ^ > j cos-B s^t 0  (03.114-)  280 A3.32 Equation A 3 . I l l can now be expanded as V^  1  -  2^0-co zt) 5  + t+fiofa  4 / 5 ^ , 0 -CP52t)  + 6  4.  .cos  9  D  s^t  ^|/3a^,  +  + / } , ^  ^  CPsPoff^t  +  4 £'  When the secular term i n Equation 4.61 i s neglected (see the d i s c u s s i o n on page A3.27 ) we have:  $  -  and consequently  =  2  t  Zb  -  4-s;~b  (A 3. H o )  A f t e r s u b s t i t u t i n g f o r 3 , sin(.t+6) and y from Equations A3.100, A3.103 and A3.113, the expansion of 2 v v 5  g  can be w r i t t e n :  1L s ^ l b - 4 j ; ^ b ^ - 4 t o s 9 | 4 - - J l /  +e  0  o  c o b -4-oo"2.fc +_Mco.?3 S  5]  281 A3.33 4- £ (A l. u?)  When evaluating the q u a n t i t i e s Vg and 2VgVg, the reduced form of and Vg('t) must be used (see page A3.27).  Vg(t)  Apart from t h i s c o n s i d e r a t i o n ,  the remaining terms i n Equation A3.94 are r e a d i l y evaluated, and from Equations 4 . 6 1 , A3.100, A3.103 and 4.84 we o b t a i n :  "4S  lV V $  g  -  u_ c o t - ScsaZfc  •/*,r«-/*0  + 11 c o 3 t  Cs>l  - iii  c» 4 t  CA3-H8)  (A3.\n)  For the product u^v^:  U. \/ s  S  = £ 2.^-^SecoCt+p) J j 2^}<cCk+P) +  which can be developed as  From Equations A3.102 and A3.103:  (A3.no )  282 A3.34 and also  2±y  e*,tfo  +  ^ ° < ^ ° ]J +  e\*,^r, +  (*3-*3)  3  (see Equations A3.98 and A3.113).  From Equations A3.103 and A3.104 f o r  sin(t+9) and cos(t+9) we obtain  s u t ( t + 8>c©s(t + e )  and t h e r e f o r e ,  -  e  j. s*ib  =  e 9.c^Zt 3  4-  +  (/ij a t )  OfeO  using Equation A3.101:  * ^/s,*" + 2 /*<»/V ^  b  -  e3 £ z  s-zfc  s  +  2^  X 0  0  3  <*>zbJ  + oCfeO  The f i n a l term i n Equation A3.121 can be w r i t t e n  - e  5  where 3 , cos(t+9) and y are given i n Equations A3.100, A3.104 and A3.113.  When the reduced forms of v , v^ and v 5  ing terms i n Equation A3.95 take the form:  g  are used, the remain-  283 A3.35  n> c<nQo | II 5 « t - 2s:«.2b + i i sc«.3t^ i ifc It J  (A 3.ix?)  D  1  Jkrab - GoJU: - 1  ' *1>4 4  £7 s ~ b - 4 2 sCv2.fc + W  s  8  L  [  32. 3 :  I  *.W3b - J i ;~4-b j  bk  bit  32. 32.  J  s  sc~2.b + j £ $^.A-b M-  I T '  e  -+- O ^ )  J  M-  (A*.1*1)  (A*.r&o)  ( A 3 . 13-1)  ]  28^  A3.36 We are now in a p o s i t i o n to assemble the various terms i n 2 2 Equations A3.92, A3.93 and A3.94.  The r e s u l t i n g expansions f o r e u ,  2 2 2 e v and e uv are given below.  4of,'  -  1  -  Zp,(l-fO^  £ I - 3^ cob + toJ-t-.1 c^>3b ^  (  -t  —  otto i,  t- •*- t » <"-U  —  Cao > i  13  ZfS' ~ l  t- e  ^to5 |l-3 0  + c^Zt- - ±  to  3b ^  + 4 p , ( l - fO<*-, f ^b-<*>2.fc -  285 A3.37  3  Zp/l-fO  +  p c ,9 >0  er  c>  j  2^,0.-^0  ^ 4 - ii. c*>b - 4  J i si«.ib  felt  1  -t- JL c o 3 b  ^  (A3.»33)  -4s«.b?  Ca b - S c < o 2 . t  -t- j | c o 3 f c  -  I?-] Co 4 b 7  2. 13  siwZb — 4 s:»vt  -  2/*,(l-/0  f e*>9 x  0  3  £ A- -JL <vot - 4 <*>ab +.  - 3_ 2-  Xb -  \±  ^3b-^  is o o 4 b ^ J6 J  -4- . 0 ^ * )  286 A3.38 (i  0  5  .+ p y  ;^t  o  o  coPoCeot  +6  +fi^-fO  - 1 « . ~ Z b -t- i i s ^ n ^  ^ 11 ««-Zb-gsiwt ^ - ^ 5 , c o P | Jl 0  (h4.  •+ p C\-p _') x  l  7  L  5 7JJ- s;«.b - 43 s « 2 b +• xj $ w3b - Ji ; ~ 4 t " £ L s & s xtJ t  b -2«;_x.b 4- Ji s^-ib ^  + /3^<ic9 ^ s ^ b - Z s ^ Z b + 5Uv3b 0  L  i'*4)  I- 3 Z  b^t-  ^  s  - ^  5 Zcob- c o l t - i  j  i - 4-  *>*-  64-  J  287 A3.39 The nonlinear functions  and  can now be evaluated f o r a s p e c i f i c order i n e when they are r e q u i r e d . A f t e r t h i s necessary,  but lengthy, d i g r e s s i o n , we return to  the a n a l y s i s of Equations A3.22 and A3.23 f o r u ^ and v ^ .  From Appendix I.  Equations A l . 8 and A l . 9 :  CA3.I3S)  (fl3.'30  ^ OCi>  Now s i m i l a r terms also appear i n the expressions f o r E-| » F 2  F  1 3  (see pages A1.4 to A 1 . 6 ) , 1  S  U.,-  =  '  E  13  a n d  o i t w i l l be expedient to e v a l u a t e , i n  a d d i t i o n , the c o n t r i b u t i o n s 0(e) and 0 ( e ) A3.136.  1 2  from Equations A3.135 and  We have, from Equations 4 . 2 5 , A3.98 and A3.104: £. 2erf, -  CA3-I3?")  (A3.I3S)  Note, i n c i d e n t a l l y , t h a t ' t h e complete form of v ( t ) g  i s used here, i n  contrast to the reduced version given by Equation A3.92. elementary manipulation:  A f t e r some  288 A3.40  r -€  1  $  p <^>9 ^ i  0  ]  3^,«<o^  (l -3coZk)  ~  I  px^e*  + 6  V  tot, c o z t  ^ot,- +  =  ^ 4oP 0  ]  ^  0  ^ 5j^b-3  s ;  ^3b"^  C v-3c«Xb')  +  + 6  s;«.Xb  =  - 2-  tio^o |  s^b+  «c*-3k  ^  X  I ]  ]  289 A3.41. From these equations we obtain d i r e c t l y :  2-  The nonlinear functions can be evaluated from Equations A3.132 to A3.134.  A f t e r some rearrangement, the nonlinear terms may be expressed  i n the f o l l o w i n g form:  It  290 A3.42 General expressions f o r the q u a n t i t i e s -2 -  -2h>j£s  .  + a_u LdbSV at J  a r e  5  £jf-s  -  ^fs  Ibtbv  and  bV J  given i n Equations A3.28 and A3.29, but at t h i s stage  of the development Equations A3.30 and A3.31 are no longer adequate. now know that | | = 0(e ),  § £ = 0(e ),  5  6 = 6  Q  + 0(e ).  | * + 0(e ),  5  5  | | = 0(e )  We  and  3  With t h i s value of 8, equations A3.14 and A3.15 can be  3  revised so that a^'C e*9y Acos C  +  -*• •••  )  • )  =  e  =  1  %  0  3  ^  +•  OC «)  .  +'e*  +  £  fe 0 s  oC**>  s  CA3.IV?)  (A?.UtS)  Consequently Equations A3.16 and A3.17 take the f o l l o w i n g form:  since  Hv  sine  Q  = 0  (see Equation 4 . 7 8 ) , and we can w r i t e  ' ar  av  t j r  ^ J (fl3.i5i )  - (i.h*2* - AcQ+Zjj -fi, * 3£J \ 9  (. " ^ r  ' ar  t-  atr  *6*j/3„<^*.+ L  a  av  •  '  ar  /  av 5  -•  •  .  +  air J  aV  [_  u  .  aV  ^12? \to  6 +  a*c  at  0  Oft')  av j  '  '  ,  <>vj  .(fl3.'J2">  291 A3.43 where Equations A3.151 A3.30 and A3.31.  and A3.152 are the revised form of Equations  From Equations A3.28 and A3.29, and using the above  expressions:  d U5  -%  — is J  -  6b ^ O i y  -Z  .b  L ab«?r  {' ix  iv  i r  1  -  i v S  t*>® df>  2  0  s  ar  Cost  2V  In a d d i t i o n to the o r d e r r e l a t i o n s h i p s given on page A3.42. we have 3V= 0(e ),  = 0(e ),  5  tly'  5  at"  and  £r = 0(e ) 5  at  t l y (from Equations A3.34 and A3.35)  1  fu* at*  and  and  [*aV  = 0(e ). 5  at* 3V at S  Consequen-  are both 0(e ),  2  so that  From the expansions of  db  L  iv-_  given i n Equations  A3.69 and A3.70, and with 8 = 0 (see Equation A 3 . 8 3 ) , i t f o l l o w s that 2  1  <^ dVj  ^b 1 <h  x  ^b J / , V >  L  0  +-  c)u  5b  5  -  era  '  292  A3.44  With the order r e l a t i o n s h i p s given immediately above,  together  with those on page A3.42, none of the expressions defined by Equations A3.74 to A3.76 has any c o n t r i b u t i o n at 0(e) and  A3.23.  or 0(e ) i n Equations A3.22  There i s , s i m i l a r l y , no c o n t r i b u t i o n from terms i n v o l v i n g  the d e r i v a t i v e s of Ug, Vg, Uy and v^ (see Equations 4.55,  4.56 and 4.69),  A f t e r d i f f e r e n t i a t i n g Equations 4.83 and 4.84 f o r u ( t ) g  and  V g ( t ) , we o b t a i n  if-  Zi + _5_ ts^fc £ 64-  bb  fO^ if-  j" ~3_ s;^k  + I3S Co 3b 64-  + WHS s»\v3.b  bb  A l s o , from Equations 4.25 and 4.26 f o r Ug and v ^ :  =  p> c^> ( b + 6 0  (A3. 16 o)  <)b^  .1-  a_v  - 2/b J<«-(b + P ")  5  and  t h e r e f o r e , using the expansions of sine and cose i n Equations A3.149  and  A3.150: cos b  b_U$  (A.3.lfrl)  bt  x  aS/  s  ^  - 2 c o 0 J * /s d  0  +• e * A I x  s;^.b  +  o  (A3. 143^  293 A3.45 The non-homogeneous terms i n Equations A3.22 and A3.23, which we can now accumulate, are given below.  4-  L  a^  o - / o  8  at?  a^  3  5  2  L  Cft3.ifc'5).  at  I  av  a IT  av  j  3s.  29k  . A3.46  <*3  p fb c^9 x  0  -  0  us  /i 0 + l  When ctg s a t i s f i e s Equation 4.82, - . i l / t t C l + w,)^ ( i - / x . ^ ) 4  2-  /  t  |  . W - / O  the two terms  X  5C«. 3 t  6( 1 + j O  *~°i-  3  and  i n Equation A3.164 cancel (see also page A3.15  f o r a s i m i l a r r e s u l t concerning a-j), which s i m p l i f i e s the expression f o r E* s l i g h t l y . 2 I f no term i n v o l v i n g t  i s to appear i n v ^ ( t ) , then from  Equation A 3 . 9 :  Now (from Equation 4.58)  and consequently Equation A3.166 s i m p l i f i e s so that  av which j u s t i f i e s the assumption made i n the d e r i v a t i o n of Equation A3.89. To e l i m i n a t e t s i n t and t c o s t terms from u-j~|(t) and v . ^ ( t ) :  .  295 A3.47  32-  From Equation A3.168, 3g(x) w i l l v i o l a t e the condition f o r uniform v a l i d i t y i n x unless 9 ^ i s zero, so we have 6>  -  3  Chz.no)  o  -  o.  (A3.17-0  When 6 ^ i s constant, Equation A3.53 i s v a l i d and we can l e g i t i m a t e l y use the s p e c i f i e d value of 3 , .  The term  1  i n Equation A3.169 i s now  oX  z e r o , and a f t e r some manipulation we can w r i t e  tO* £  p  Note that cos 9  N  >4j (i+p.r*-  = 1, so that  u  4.81). of  +• \->s  a  '  ?  cosB = H a  1 6  =  u,  / c  '~  0+  o  (A3.  .  \z  (see  Equation  A l l the terms i n Equation A3.172, with the exception at present  sL©5 ,  are constant (see Equations A3.50 and A 3 . 8 6 ) , and consequently  d©5  - constant  (A3.t?3)  296 A3.48 I f we also assume that  and 6^ are constant:  sec 9,  (ft 3.174  (see Equations A3.211 and A3.244). From Equation A 3 . 9 , one secula r term i n v ^ ( t )  i s generated by  the expression  I (>-f J)  1  6  -J  ^-  3  We have, from Equation A3.169:  -t  13-oi,  and when p coiB )  0  - 2^,0-^)  i t f o l l o w s that  (A3.(?S)  (see Equation A3.53), so that the s e c u l a r c o n t r i b u t i o n . c a n be w r i t t e n  )  297 A3.49  The remaining secular terms i n -]](t) r e s u l t from q u a n t i t i e s i n Equation v  A3.165 which involve s i n 2 t and s i n 3 t (see Equation A3.11).  Note, however,  that by Equation A3.175 the c o e f f i c i e n t of s i n 2 t i n Equation A3.165 i s zero.  Consequently, i f we assume that ag i s constant (see equation A3.208)  then to cancel the term  -e (3tag) 6  in  v (t): 5  (A 3 . i ? 0  As u s u a l , we take  -which leads to the f o l l o w i n g value o f  SfjL,(  | _ fO^"  [  l < ?  +  ^ 3 ^ , ]  -  CA3.IT8)  The functions Ug(t) and Vg(t) can now be evaluated, since the values o f 0-j, 6  2  and 8  3  are known.  c o n s i d e r a b l y , and we then have  Equations A3.42 and A3.43 s i m p l i f y  298 A3.50 Non-homogeneous s o l u t i o n s corresponding to these functions can r e a d i l y be derived from Equations A3.10 and A 3 . 1 1 , and when the secular term i s included ( c f .  page A3.17) u ( t ) g  and v ( t ) g  +  take the f o l l o w i n g form:  vsb  6±  -  6  \±  Cos,  (A3.(61 >  2-k  6  £ 2 s*  3  Xb  +  XI  b I  (A3.ISO  Next we consider the s o l u t i o n s of Equations A3.24 and A3.25 f o r u.j2 and v - ^ .  From Equations A1.8 and A1.9 i n Appendix I :  '7-S (to3tr  6A-  '  +  i»9  3|5  ^Sfc-  (A3-lS3^  1-2.8  i-  JS si^t 64-  +  IOS ; ^ 3 b is-S S  -+  ?rS <c Sb ra-S  CA3.I24-)  H  Solutions f o r Ug(t) and Vg(t) are given on page 4.15,  and a f t e r some  manipulation we o b t a i n : H,A 6  '  3<*>Zb) -  1  CA3-I85)  3V. <^.xt •  /6  i6  4-  299  C 6  •A  I -  "3  Ccr> 2 - 0 «» -  A3.51  3 u . , S<«-2b  8  (A3.I86)  4-  /6.  From Equations A3.139 to A3.142, the c o n t r i b u t i o n s i n v o l v i n g u^ and v^ can be w r i t t e n : 2b^  - 3v,  -2.1  }  (f\3.lg?)  Oft)  £ 6  +£<^ab  - il c ^ t  +3<o>3t  +  2-  ^ , Cl- 3  + -M-.CI-  /O  2.  IS_  tS  i -  2-b  J \  c n i t l  (A3.I88)  ?  Equations A3.183 and A3.184 can now be evaluated, and we have:  96  7£  4  cob  + ]75 cr>3t  64-  i5 64/  J6  p.^i  Cfl3.i81)  +  ioS iaS  3Z  +  3i£ <v»Sb  l>8  t3.8  t i t  2  3-2.  s ^ b  +  3jj>  b  (A3.l<*o)  U S  rV2Tr Cl-  3c«2.t)  2. '  Note the c a n c e l l a t i o n of secular terms i n v o l v i n g t s i n 2 t and t c o s 2 t , which appear i n Equations A3.185 to A3.188. A f t e r some a l g e b r a i c manipulation the nonlinear functions can be  300 A3.52 arranged i n the f o l l o w i n g form:  II 5x  (A3. Ill")  [  Zfc  sew Z t  (see Equations A3.132 to A3.134). The q u a n t i t i e s  -zf^jfs  -  "1  |_atav  J  and  - 2 T i*y>  +^5*1  Lata*  ar J  oCfc ) 6  oCe )  i n Equations A3.24 and A3.25 can be determined from Equations A3.28 and A3.29, using Equations A3.151 and A3.152.  We then have:  -Z  oCe ) 6  +  Zc*o©  (  I  '  br  dv  J  6  301 A3.53  [  S A , a P +/>,iJ£f + ^ ^ 7 ^c L >v ' dx bxr S  -  x  a v/g  +  }tdx  S£  t  Tx  (ft 1  .ISO  dx  because, from Equation A3.170, 6 ^ = 0. At present the d e r i v a t i v e s o f a , B and y with respect to T are all  0(e ),  |^-=0(e ),  6  4  ^9 = 0(e ) and 6 = 9 + 0 ( e ) . 6  A3.34 and A3.35 i t f o l l o w s that  $j±_s  and  ^y_s  From Equations  4  n  are  0 ( e ) , and 6  consequently  =. o dx* _  OCeO  With 2 9  =  9  3  =  °'  E <  l  u a t  i°  n s  A3.69 and A3.70 s i m p l i f y c o n s i d e r a b l y ,  and we o b t a i n :  L  at 1  at  J (p^*)  COS  OCe*) None of the q u a n t i t i e s defined i n Equations A3.74 to A3.76 2  3  c o n t r i b u t e to Equations A3.24 and A3.25 at 0 ( e ) or 0 ( e ) .  The remaining  non-zero terms i n Equations A3.24 and A3.25 are given below. From Equations A3.181 and A3.182 f o r u ( t ) and v ( t ) : g  g  302 A3.54 (A3.119) •  at"  1  '<Jb .  L 6  (A3.  3  I l l )  From Equations A3.162 and A3.163: cofi(, c o t  •2/3, e o P  0  (A3.  *-.t  2O0)  (A3.  From Equations 4.60 and 4.61 f o r Ug(t) and Vg(t)  =  piC'-/oF  4t<olt  C » - 4 - s u v t  -2<uot  -  ]  (As.  J i ^ Z b J  XOTL)  (A3.-3.03)  and t h e r e f o r e , using Equation A3.53 f o r B^COSOQ:  cn  (A3- ze>4-)  It  aV at*  (AS.ZoS)  We are now i n a p o s i t i o n to assemble the non-homogeneous terms i n Equations A3.24 and A3.25. E* and F* then take the form:  768  *>r  1  ar  J  ('-^O  J  303 A3.55 +  cot  £'  a*"  '  ar  ' a v  J  6^-  - OS  3•  -j**.") fco3fc 13.8  2- Sot  t  •+ sovcb  r  **  <?v  a*p"  J  £4-  C -ro  aV  (j  17-S  -  r  o  4-  3Z  Secular terms i n v o l v i n g t  are e l i m i n a t e d from v - ^ t )  if  (A1.208)  so that equation A3.178 f o r a and t c o s t from u ^ and v - ^ :  g  is valid.  To remove terms c o n t a i n i n g t s i n t  3014. A3.56  *-  '  air  From Equation A3.209, i f 9  d~c  4  J  i s not zero 3g(x)  v i o l a t e s the c o n d i t i o n f o r  uniform v a l i d i t y i n x, and therefore  ^4  -  Now when 6  (A-i.ait)  °  4  = 0, we obtain from Equation A3.85:  i n which case Equation A3.210 can be w r i t t e n  'A-. Iff. / « , 0 - / O  -+  '£L K . ^ f O ^ - j O  =o  V  64  From Equations A3.84 and A3.173 both 3. and 4  3_f ar  5  are constant, so that  305 A3.57 SSfc at?  =  constant  .  (A3.2IS }  I f we now assume that e  5  and e  6M  b  g  are constant, then from Equation A3.214:  +  105 ( i - - f O ~  —.  sec 9  (A3.lib)  r  s  i« ( n- JJ. , ) The quantity - 3  l'  av  '  J  64-  3  - z ?6g  a-rr which generates a secular term i n v - j ( t ) , 2  can be s i m p l i f i e d using Equation  A3.214, so that the secular c o n t r i b u t i o n takes the form  [  fa.  3S43S4-  32-  .0.-/0  ~  A d d i t i o n a l secular terms in v-^Ct) r e s u l t from those expressions i n Equation A3.207 which contain s i n 2 t , s i n 3 t , s i n 4 t and s i n 5 t Equation A3.11).  (see  The combination of these s e c u l a r q u a n t i t i e s must  e l i m i n a t e the term -e^(3tay) i n V g ( t ) , and t h e r e f o r e :  4  We t a k e , as before,  v*  (A-s.ai? )  306 A3.58  and  if  i s assumed to be constant (see Equation A3.240), from Equation  A3.217 i t f o l l o w s that i X f 0+|<) + ££ Yi + t  ix-  64-  3£  (A3.2l<0  Now that the values of ^ to evaluate u ^ ( t ) and v ^ ( t ) .  a n  d 6^ are known, we are i n a p o s i t i o n  Equations A3.77 and A3.78 can be w r i t t e n  i n the reduced form:  <Vo2-t  +•  3f  p,(i~pj)  Cr>A-b ( A 3 . 2 X O )  16  0 -pO  ^ and  TT^o  8  from Equations A3.8 to A3.11 u  ' 1 Q  6  ( t ) and v-|g(t) can therefore be  derived as: fl (l-ftj.^ t  L  8  s-ia-Ci+jJ.)  J  37 +  6 7 S ^t,  3a  cob  307 A3.59  - i  fr3 s^A-b L  -  3  (  s^.b)  b -  -  64-  +  X-zS fi,  SixCl-t- u,)  ^  4-S  If.  0  J  (fl5  ( V c o b )  .«-3)  0-/**)  Note that secular terms i n v o l v i n g t are eliminated i n the complete s o l u t i o n f o r ev(t) when  s a t i s f i e s Equation A3.89.  We c o n s i d e r , f i n a l l y , the s o l u t i o n of Equations A3.26 and A3.27 f o r u-j2 and v-jg.  The functions Uy(t) and v^(t) are both z e r o , so that  from Equations A1.8 and A1.9 we have - u,(l - fO  'IS  |  f  I  '  1 <»k + l l c « 3 b I  L «  e  J  - V  5  J  L  s^2-b?  Now U  r - p e 9 Cob  s  >0  tr>  0  N/ = 2y3„ Co0 s~b &  o  •+  O f t )  + y  0  + oCe^> .',  3_ s . v b +• *5 s ;  *  4-  5C^.3b  ( A 3 , aa-S)  308  A3.60 and  consequently:  1 tot +•  .3  5  4  3_ ;~b +  to?t  s  4  « t t if sc^3b 4-  0  Csp >x  0  if 8  +  i f 5J«.3b  4  6 co 3-t - if <<o4k «  (A3.1Z6  )  —  -  [  -y  0  &  £,•»,  c  _  9  1 tot - l£ < o 3 t 4 4-  I?  [" 2. c ^ b - if c*>3b [_ A4  Because 3 and  V  2  -  if  (A3.Z2>)  j;^4k  8  = 0 (see'Equation A3.213) Equations A3.139 to A3.142 s i m p l i f y ,  for E^'and =  si^Zfc  e>Ci>  we therefore obtain  p / ' - f O - j /3«,<«'Po| l l  +  feCeoXb  -  If  Ceo  4bJ  •*•  jTo ^ 1.  Si-b  + is j . - ^ t j  2L  = - y , ( l - ^  D  c ^ 9  0  2 7 si^ab - i f s ^ 4 b |  + y  e  £ 3 e ^ k - IS «to Sbj  ^  CA». 2.3-1)  From Equations A3.132 to A3.134, a f t e r some manipulation the nonlinear c o n t r i b u t i o n to Equations A3.26 and A3.27 can be w r i t t e n as:  3x  '  ^  309 A3.61 .  4-  CA3.230)  [i lit CA3 . 23\^) 64  8  For the expressions  •  -2 | d "-s |_ d t a r  ^ s l  and - 2  J  }L?S  + ^-s ,  we have, using Equations A3.151 and A3.152:  it7r  -2.  +  L  ;>k3 tatr  X t)/3  ix  ar J QCtT) ?  ceoPp s ^ t  +  >x  2&9 f 0  A bB? 0  ( A 3 . 2.32.)  1-/3,^6 "I  ^st  2 a*,. atr  ( A - i .133)  CoP„ cost ,  2  •tr  310 A3.62 since $  2  =  9  4  =  0  The d e r i v a t i v e s o f a , 8 and y with respect to T a r e , at t h i s stage of the a n a l y s i s , a l l 0 ( e ) , | ^ = 0 ( e ) , 7  5  $9 = 0 ( e ) and 6 = e_ + 0( 7  (see Equations A3.208, A3.211, A3.212, A3.215 and A3.218). the q u a n t i t i e s  v  z  3_!f5 and a_v>  It*-  bX*-  Consequently  5 do not contribute a t 0(e ) to Equations  A3.26 and A3.27. I t i s necessary to extend equations A3.69 and A3.70 to include terms 0 ( e ) , but as 9^ = 5  = 9g =  = 0  t h i s presents no d i f f i c u l t y ,  -and we o b t a i n :  I  a t *'  at  j a_y + h± c Lafat J s  5  There i s no c o n t r i b u t i o n , e i t h e r at 0 ( e ) o r 0 ( e ) , from the expressions 4  3  defined i n Equations A3.74 to A3.76, and f o r the remaining non-zero terms i n Equations A3.26 and A3.27 we have:  dj*  to  at'  - a*, at  /-",(<-r  3- V-3.Z5  p  , ?  tost  +  f  + 2 i l cos4t 64-  +  3^  [ X - .e^-b 1  -f- -45 j< , 6 - j O  -  p., coj? 2.t S  T3 + 6 ? £ ^ _  L  64-  2o s ^ t  7  S-ia-Ci^p.) J  (A3.a34)  311 A3.63  -  + ^,0-/0*  - 3 ^ s u l :  f ^.  4-  za-g'/x,  -  4-5 ^ . Q - f Q y  ?-5^b -'/i!>  e  to 1  CA3.23?-)  5  + 16  The non-homogeneous terms i n Equations A3.26 and A3.27, which we can now assemble, take the f o l l o w i n g form: - 6 t ^ , . ' H .  a v  }Xr  _ 3 _ f / 4 . / i  3  + ^ 3 * " ^ ]  f'l-^,^  -  p/'-fO/*  +  64-  J  1  3  8.  2  •+ «o b  •'+11 (*,yx^«  + 75 su-3fc  •+ cos4b  r  *5 j « . & - f O ' / v * 0 . - + i i i p / ' + f O ' ^ ' - r O 64-  31  Z  3 *  3o£> -t-S^b (.' a r  <ar  J  3  CR3.2.38)  312 A3.64  - <vob  -  ,  - s;».zb  *  64  L 4-  g  2  I f no term i n v o l v i n g t  i s to appear i n v  1 3  (t),  then from  Equation A 3 . 9 : ^ ?  =  O  Cfl3.Z4o^  ,  ar and therefore Equation  A3.219  is v a l i d .  Terms containing t s i n t and  * c o s t are eliminated from u-j^ and v - ^ when:  a*  it  t  ar  ar  Although i t i s not required i n the present a n a l y s i s , i n p r a c t i c e we must take Y Q = 0.  The reason f o r t h i s choice i s discussed i n Chapter 4  (on page 4.22 ).  With t h i s value f o r y  n  we then have:  313 A3.65  Unless 0g i s z e r o , B^CT) w i l l v i o l a t e the requirement of uniform v a l i d i t y i n x , and consequently  Zfe  ~  O  ( A » . * « )  .  Now that 6g i s known to be constant, Equation A3.174 i s v a l i d , and the s p e c i f i e d value of 8 3 can l e g i t i m a t e l y be used i n the a n a l y s i s . Equation A3.215  -  From  i s constant, and therefore using Equation A3.242:  constant  .  fjnW)  at A l s o , i f we assume that both 6  g  and Qj are constant,  sec 9  Q  then  .  -(A3.24-?-)  o  The secular term i n v ( t ) must e l i m i n a t e the q u a n t i t y -e 1 3  (3ta ) g  i n v ( t ) , and from Equations A3.9 and A3.11 we then have: 5  +  W  }  -  By the usual argument:  C jr.^*^.^]  .-  ^ia>  v  3  *  8  ( A * . * * * )  3114A3.66  and, assuming ctg i s constant, from Equations  4.81  and  A3.174  f o r 3Q and  $ 2 we o b t a i n : 0+ j*,)*" ( ~ / O  ifj  l  pi  +  -|  2-175  4 - 2.0 25 p.,  (  (A3. 2-So )  where Y Q i s included to f a c i l i t a t e Equations  A3.89,  A 3 . 1 7 8  This completes solutions for ^'(t) Chapter 4 .  and  comparison with e a r l i e r r e s u l t s  (see  A 3 . 2 1 9 ) .  the general a n a l y s i s of 3-body motion, although  and r ^ U ) derived from these r e s u l t s are given i n  315 A4.1  APPENDIX IV — VARIATIONAL ORBITS IN THE RESTRICTED PROBLEM  We are concerned here with the homogeneous and non-homogeneous s o l u t i o n s of Equations 5.23 and 5.24,  which define v a r i a t i o n a l motion  about the f i v e e q u i l i b r i u m points of the r e s t r i c t e d problem.  For the non-  homogeneous s o l u t i o n s i t i s necessary to consider motion near the c o l l i n e a r points separately from that near the e q u i l a t e r a l t r i a n g l e p o i n t s .  The  present a n a l y s i s , however, i s l i m i t e d to the i n v e s t i g a t i o n of motion near the e q u i l a t e r a l t r i a n g l e p o i n t s , so non-homogeneous s o l u t i o n s f o r the c o l l i n e a r case are omitted.  A4.1  The Homogeneous S o l u t i o n I f we take x =x-j,  x = x , y = x 2  3  and y = x^, then  (neglecting  the non-homogeneous terms) equations 5.23 and 5.24 can be w r i t t e n i n the form  3c = A  (A 4.  0  Here x = (x-j, x , x^, x ^ ) ^ , and the matrix A i s given by 2  A  o  I  "°. o •o.  o o -z  o  o  2. 0  i  O  ( A4. % )  316 A4.2 Taking the Laplace transform o f Equation A4.1 we obtain  ( A 4 . 3)  where I denotes the u n i t m a t r i x .  From Equation A4.2:  and i f we now define  then Equation A4.4 can be w r i t t e n  deb ( s i - A )  =  r - + 2-X, r 2  +  x  (A4.6)  z  where (A4.7)  (A4.g)  From Equation A4.6 we therefore - have  deb C s l - f l ) = j s V X , +  A, - ( A * - \ ) ^  A f t e r some manipulation the f o l l o w i n g r e s u l t can be obtained:  (A4.<0  317 A4.3 (si-A)  —i  C  3K  ( A 4 . lo^)  <Lefc ( s I - A )  where C denotes the matrix:  2s-u,  20,-o^s  s -*- o. 1  Note that the q u a n t i t i e s £^ and  and  depend on the p a r t i c u l a r values of  ( c f . E q u a t i o n s 5.20, 5.21 and 5.22 on page 5.6 ). « • n c  -  - •  -  HUH ICOOI  ...  u u i  — J. J u  i . -I  o i,ci  i o  i u i  ... i  u O  .!. I. -  - ~ . . J  o n e  c i { u i  T - J. _  T  IUICIUI _  ^ _ ... _ T Ci  _  i c i n y i c  ~> — — ,~ J - — i  m , o  and 1_ (see Figure 3-2 on page 3 . 1 8 ) , f o r which 5  (_A4. i z )  2,  yr/  - 1  L2>  L7-  Note, a l s o , that  ^3 = 1  +  5.21 and 5.22: o, - —  '  -  K  -  k, z  2  at  L  Y  at  L  C  (04.14-)  In t h i s case we have, from Equations 5.20,  318 A4.4 at L : 4  3L  and at L^:  2.  From Equations A4.7 and A4.8 i t f o l l o w s that \  =  V  and  1' + k  0  + K  (A4.n)  = (l-^o)'' - * k / l - k ) 0  + 3 1 ^ , 0 - / 0  (A4.ZO)  consequently  \  x  - \  x  =  A-C^+O  + k, " - 3 k ^ J i 7  r  , _ ) .  Note that Equations A4.19 and A4.20 are v a l i d both at  (A4.20  and L g .  The character of the homogeneous s o l u t i o n i s determined by Equation A4.20, and bounded motion only o c c u r s , i n g e n e r a l , when  4(k -rk,) 0  +  k,- - S k ^ j ^ f i ' / O 2  >  r«4.ix)  O.  I t i s possible f o r bounded motion to take place when Equation A4.21 not s a t i s f i e d , but only f o r s p e c i f i c i n i t i a l c o n d i t i o n s .  is  We now r e s t r i c t  the s o l u t i o n f u r t h e r to those cases f o r which Equation A4.21 i s  valid,  319 A4.5 which implies that >  - _L j - 4- C i< + k, >  i  e  "5  or,  k  alternatively "7 J -  <  j_ -  -J  2-  since u  11-  3 k  ^ VJ3 requires that u ^ 0 - 5 -  2  2  I f the constants kg and k-j corres-  pond to a conventional binomial expansion, then kg = 1 and k-j = - 3 / 2 . Equation A4.23 can then be w r i t t e n  * it  1  - (r,)'  1  which i s i n agreement with the value given i n Szebehely [1  19  ].  Taking the inverse Laplace transform of Equation A4.3 we o b t a i n , using Equation A 4 . l l . : • Cb^  -  C05  co  =  s  fc  si*. u>< b Zo^Tcto^  +  (oi^-O^  £Co"} + 20^ LJCO } + 0 ^ <j<o) -  320 A4.6 +  Cos  i(fc)  k  -  -  s^i^ b s  2 0 , 3t£o">  -  o^ito)  4- 2 0 ^ 3 ( 0 " )  -(-  ( o , - ^  7  +  (o,-^  -  - ^ y(o)  to.  +  COS o><t  +  Sit tJ t  -  cos ojt  u  2 0 , =c(o")  -  0^ i C o " ^  + ZOa. LJCO")  0  ) 3(0)  (A4.1?)  where (ft4.ZS)  I t i s more convenient, however, to express x ( t ) and y ( t ) i n the f o l l o w i n g form: CA4.3o)  (04.31)  where a , b , 9 and <>j are a r b i t r a r y constants. Q ,Q - , 6* and <j>* are defined below:  The remaining q u a n t i t i e s  321 A4.7  Q  =  u  [  4-oJ-  -»- 0? ] "  (A4.33)  ^  (A4.34)  Note the symmetry between expressions associated with the s h o r t - p e r i o d (u> ) and long-period (a^) terms which i s revealed by t h i s f o r m u l a t i o n . s  A4.2  Non-homogeneous Solutions The non-homogeneous form of the v a r i a t i o n a l equations can be  written £  - 2 ^  +o,x  + -O^tj  =  E * ( t )  y  + Z i  +  t o ^  »  F*Ct)  3  (ft4.36 ) >  .  (44.3?)  I f we s e l e c t the vector B so that B = (0, E * , 0, F * ) , then Equations T  A4.36 and A4.37 can be expressed as  S  =  f\5c +  5  ,  CA4-36)  322 A4.8 . where the matrix A i s defined i n Equation A 4 . 2 . Taking the Laplace transform of Equation A4.38:  5L( ^  =  S  (sI-A)  _ ,  S C o ) + ( s l - f l ) " ' ZCs)  ( f t * . 3*0  and the non-homogeneous s o l u t i o n can consequently be determined from Equations A4.10 and A 4 . l l  i f B(s) i s known.  When E*(t) and F * ( t ) contain  terms of the form s i n u t and cosoot, three d i s t i n c t cases are p o s s i b l e :  f  co f  1.  w  2.  w = oo  3.  oo = co^ .  s  s  u  L  ;  ;  These are considered separately i n the a n a l y s i s that f o l l o w s .  A4.2.1  Non-resonant case When co f w f co , E*(t) and F * ( t ) can be w r i t t e n i n the general s  L  form £ * ( b )  =  F *  =  Cf)  C<„  /S  0  and we therefore  2»  +. Ct, Swvtofc-  -t  /S, sf^tJb  +  +  /J,.  cx>5 t J t  have  sHiO  1 -  '.  o L , . cos o b  S ^ + O *-  ,  .  (fl+**o)  . ( A « r . <*i)  323 A4.9 B(s) i s now known (from Equations A4.42 and A4.43), and the nonhomogeneous s o l u t i o n may be.derived from Equations A4.10 and A 4 . l l as the inverse transform of ( s ! - A ) B ( s ) . _1  A f t e r some manipulation we o b t a i n :  A  2  UJJ- ( toot, »2|Q  +  + •oCft.Q.,.-*^)  2/3,  + cos o b t  ^  — cx>s tob  — SiVv u><- b  (  CO*-  (A*.44-)  32k A4.10  — Cos t->jt-  —  —  2-ot,  StK-0>fc  - to$ cob I  ( co - O L  -+ c i O x  (w *-  W" ) 1  s  A4.2.2  -V 2-coot,  x  Resonances i n v o l v i n g s h o r t - p e r i o d terms  For the second case, when co = to , E*(t) and F*(t) can be s expressed i n the f o l l o w i n g general form:  F*  Ct)  =  '3".**.o,b  . +  S", w s c j j b  t  (A4.A-?)  325 A4.ll. so that  ( A4.  The corresponding non-homogeneous s o l u t i o n of Equation A4.38 i s given below.  MH  — Siw-lO.fc  to.  - cos to. b  t Si«- to-b Zto ( s  -  t cos (O b s  to^-cojO  4-  2^,05*1  (A4, 5-o)  326 A4.12  NH  COS  t  CoS«0, t  tcc-5 ^  A4.2.3  t  (fi4.Sl)  Resonances i n v o l v i n g long-period terms When to = to^ we have:  (A4.Sz) (A4.53)  327 A4.13 and therefore  The non-homogeneous s o l u t i o n s can, i n f a c t , be obtained d i r e c t l y from Equations A4.50 and A4.51 by interchanging w and o ^ . s  ?  2  that i n t h i s case (to (a£ - tto£) transforms to - ( t o s change. We then o b t a i n ; v  - cos t ^ k  0^  e  VZJ,  2 s  Note, however,  2 - w ) a f t e r the L  inter  328 A4.14  Cb)  =  "  sew. cjj b-  ^ L S^C^-O +*-y«>l - 2jr, V J  u  - cos <o  s  11  C JJ^-«Q  t  + t  cos to.  b  + °j-}f\  V° °'T  329 A5.1  APPENDIX V — SOLUTION OF THE FOUR-BODY PROBLEM  Two r e l a t i v e l y lengthy d e r i v a t i o n s , associated with the expansion of nonlinear and d e r i v a t i v e terms r e s p e c t i v e l y , are given below i n Sections A5.1 and A 5 . 2 .  In the f i n a l section we continue the a n a l y s i s  of Chapter 5 to derive s o l u t i o n s f o r eu(t) and e v ( t ) that are uniformly v a l i d when t = 0 ( e ~ ^ ) and which define the p o s i t i o n of p^ to w i t h i n a 8 constant e r r o r 0(e ). A5.1  Expansion of the Nonlinear Terms For the function  \r d  |  we have, using Equations 5.32 and 5.33:  Uniformly v a l i d expressions f o r the motion o f p and p are given i n 2  3  Equations 4.101 to 4.104, so that 1.  r  330 A5.2 I f Equations A5.2 and A5.3 are rewritten i n the f o l l o w i n g form:  then we obtain  Now because of the l o c a t i o n of  and  (the e q u i l a t e r a l  triangle  points):  - C> ~ - •  +  =  Cfx+Sj'  + I?''  =  In t h i s case Equations A5.6 and A5.7 can be w r i t t e n  o r , more s i m p l y :  •  ..  (AS.*)  331 A5.3  43  where .  From equations A 5 . l l and A5.12 we then have the f o l l o w i n g general expansions of  '  I r  r  3  4 l |  anrl and  I  series  3  P  k  o  +  k, ^  -t-  Lr g t  v  + . ••  (A*.iO  ( c f . page 5.4), where the constant c o e f f i c i e n t s of expansion i n iuse. ( f o r i = 2,3)  0 . - * i  * =  - £) and (n. - n) can b e ' w r i t t e n  as:  " ^ 1 * 0  s  ^ - 5  The expressions  depend on the type  -  A  O  . \'.(AMI) (AS.H) (AS.**)  332 A5.4 and consequently the nonlinear terms i n Equations 5.36 and 5.37 take the form:  +  k.  + k.  (ftS.Zl)  where we have made use of the c o n d i t i o n . (A5.0.3 )  i -  which i s v a l i d at  and L ^ .  333  A5.5 Equations A5.21 and A5.22 cannot, u n f o r t u n a t e l y , be used i n t h i s compact form, and a development i n terms of e i s now necessary. expressions f o r  The  , T^, A-| and A are more r e a d i l y handled i f we take 2  IO  /•—J  «--S  A  »  «  eu- +  where e^(t)  C  1  e ^ O ^  and f ^ ( t )  +  OCe")  (As.**)  correspond to terms i n v o l v i n g e  1  i n Equations  4.101  and 4.102. From Equations A5.13, A5.14 and A5.23:  Now i f i t i s assumed that eu = O(e^) and ev = O(e^) on pages 5.13 P,  1  -  cV--  (see the d i s c u s s i o n  and 5.14),-we obtain from Equations A5.24 to A5.29: 2-fcVe  s  +  fe'°c * s  +  OCV')  (A5-3o)  33k A5.6 e V  +  L  I -  -tL  Ode")  (A«.3S).  I-  0 - / O  0  - p * ^  (A5.35>  A l s o , i f q u a n t i t i e s ( H e ) are neglected, the f o l l o w i n g approximations 11  are j u s t i f i e d :  CA5.-56)  (AS.Ifi')  -  A.  +  , —3(AS-Ao)  (AS.A-0  From Equations A5.38 to A5.41 we then have: (AS. A-*)  335 A5.7 and from Equations A5.36 and A5.37:  =  x  C V*  +  - A-^ C  P,P,_-A,^)  u  +  ( P . V A , ^  (AS.44)  The nonlinear expressions defined by Equations A5.21 and A5.22 can now be evaluated, using Equations A5.24 to A5.27 and A5.30 to A5.35. Note t h a t , when q u a n t i t i e s 0(e 2  11  2 2 ) are neglected, the terms f r ^ , g  ,  2  f  and g A make no c o n t r i b u t i o n to Equations A5.21 and A5.22. 2  some rearrangement,  After  the nonlinear functions can be w r i t t e n i n the form  given below.  + <^  2^*v ( k , + O  +  P  -  g  ^  ^  l  ^  O  +  v ^ k , ^ ^ ) ^  CAS.4t)  i-^O  L  *• • •  336  A5.8  u.k , * f 6 - £ / 0  + fe  V | k t 3k, |  u  CA5.47)  0  + 6*  0-  .  Note t h a t , although the value of ri|_ depends on the choice of Lagrange 2 p o i n t , n£  =  3/4 at both L^ and L g .  This value has been s u b s t i t u t e d  2 f o r n£ i n Equations A5.46 and A5.47. terms i n v o l v i n g e Equations 4.101  ^  s  «• '  1 0  The d i v i s o r ( l - y ) associated with 2  c o u l d , p o t e n t i a l l y , create d i f f i c u l t i e s , but from  and 4.102:  l? 8  jO*  S-K-b  ,  (*!+  and therefore as y ,  1 the terms i n question tend to z e r o .  (AS. 4--0  337 A5.9 A5.2  Asymptotic Expansion of the D e r i v a t i v e Terms From Equations 5.58 and 5.59,  u ^ ( t , T ) and V g ( t , x )  the homogeneous s o l u t i o n s f o r  are defined by:  and t h e r e f o r e :  av ^ U-s  =  Titx  — u> y  bo. s^»v  s  I bx  - u> / u  I  iP'VLf  ^tr  =  COS  S  £W s:-vEto k + ^ ] u  bX  +  bj> cos E^ut+-</>] Z bX  (A5.S3)  J  "5'  (BS -5-4) -  Corresponding r e s u l t s f o r the d e r i v a t i v e s of V g ( t , x ) inspection from Equations A5.52 to A5.54.  can be obtained by  338  A5.10 Expansions f o r sine and cose are given i n Equations A3.16 and A3.17 (see Appendix I I I ,  page A3.6 ) , and using  <O t cos 9  +  Co5u>jt « > 5 #  -  s  Cos(^ b+ 5  9)  =  coso b $  S^«O t s  si^9  (As , ss)  S<^-0  (AS. ? 6 )  -  we obtain  -s;^o k |Vv.9 5  + feD.cosB* + t ^ c o s 0 1  e  x  o  ©, *H&OJ + • - J  (As.*8)  1  Equivalent expressions f o r sin[w^t + <j>] and cos[o^t + <j>] f o l l o w from these r e s u l t s by symmetry. From Equations 5.62 to 5.65 the functions can be w r i t t e n as: ax  dv  '  $v  ,  , a-^-  and  339  A5.ll  a.b9_  [  bxr  av  av  oC O  at-  (AS.  fe  61)  bbj>  bV  ar  bX?  d~c  0Ce ) 3  and using Equations A5.57 and A5.58 we now have 3a co* ( W j k 0 )  =  co5«J b  /  s  cos 0  O  - si^o t- sc*-0 7 5  o  at I  +el  (p t +  cos  Lai?  s  •/  r  ^  1  L  ^  z-  J  }  9) a  at?  arj  cos C to b +- D ) 5  0  }  [ +  CA^.65)  0 ( V )  di> cos(to fc+^>) dY u  L a*c  - at  D  «  °  S cosO t cos 6 1 ' L  1  dv  - sc* o b si*. d> 1 J u  °  J  31+0 A5.12  {  bh,  -  cos C u> t + C^ )  -J3bV  0  u  Co. t + c O  +  o(e0  e)a si>v( u t + G ) dX s  =  0  ba. S si*.u> b c*z>0 bV I 0  J Sa, SC«- (u> 1 + 6 ) +• G, 5  I L  bx  i-  0  s  o  + co5 0 t s ^ _ E ) 7 J 0  5  cosCcO t + 0 ) O  s  "I  bxr j 9, ^ce,  +  G^cte.,,  }  cos(u> b+ 8 ) s  0  (AS. 6 5 )  }  1  +  O l V )  ar  Tx  cos ( ojc + <j> ) 0  x  (A5.^)  A5.13 cosC u> t +9)  o.  5  bX  - a c>Qo I cos u> bcos & - $i^u> t s<.v. 9 bX L J 0  a^bQo -f- a ^ 6 | 0  s  Cos(  u) l + 6>) s  I  —  '  o  o  h + I-  -  Q  s  \  £ <O fc + 0 )  - a 0, 3P  0  s  0  i f  J L  rrtC (  l-> r>  0  O  d v  3-  \  r£0 (  oX cos ((j tr -f </>) u  + 6.  1  -  0  J l  h j> ~c\i> s ^ ( o _ t + ^ > ) at? o  x  Ir  i  0  <?r  ar3  }xr  d9 5 ^ ( i o t + © ) 5  ^  a  0  d 9 I s ^ v . ^ b cos 9 0  s  &  0  dX ^ J  + tosu> \:si^Q "I 5  0  3k2 A5.14 +• e  a 0,  C o t 4-9 ) s  ar  ar  ar-  0, d0 *>r  b 3c£ Sin, ( <O t 4-  •i-  ar  0  b  ar  0  £ to 1 + £> )  0  c ^ , | s ^ ( J b COS ^ L  I  ar  + ( k L  <ar  e  ^  l<h>  a r  dr  +  0  co5 (to t+  {  9)  5  4- CoScJ^t  ar L r b. ScL 4- •!>_ Sc6.~l sCw ( u). b 4- (h.")  5  d r  + ©, a-, a_9 +• a d9 av ar  o  =  u  COS  o  0  0  s^(j> \ b  0  4-  k.  ^  j  r * c ( ,S. \- ^ A> }  "£ a r 3  £ [" b , ^ L a r  0  + b o ^ . l] a rJ j  c o 5 ( ' o b + ^. ) u  Ode ) 3  From Equations 5.62 and 5.64:  a r  (AS.?I) 7  ar  :  ar'  3if3  A5.15  ( ft  •  ar  r /  +• e \ ar a„  Cud_!f>  4-  0  oC °>  a_a  6  L aa tr ar-  ar  bl9  a r a-c  ar"  fe  4-  f a, b^9  i  ar a r  0  a  ar"  4-  a^£.  d  ar -  O^*^  (AC? 4)  1  #  and Equation A5.54 can now be expanded as:  aV  = / aV  5  av 1  L  0  ar 2  - i " 2a_ aj 1 a r ©1  t  av -1  I X 3],,, ^ I ar ar  + k  e  I  I  ar--  a r  4Sa ae  i[h,^o  t>  dt?  1  V a r /  o  ar •  0(e ) l  ar  1  e  ar  4 9,  4  0  O  )  2 S ~ ( o t 4 cf> a-c -) u  1  a r  + a.,d2iV 4 a a 2 p ,  h  S ~. (  fto  *\,  a r ^ j  -ike.  te)'I3  c  3¥i  A5.16 As mentioned on page A 5 . 9 , the d e r i v a t i v e s of v,-(t,T) can be deduced from the corresponding r e s u l t s f o r Ug(t,-r).  Note, however,  that  from Equations A4.34 and A4.35 (see Appendix IV, page A 4 . 7 ) :  dr  ar  dj>* = 14>  and  (A?.??)  c>-c  5v  therefore only the terms s i n 8 , c o s e , sincf> and cos(j) need be a l t e r e d Q  Q  0  0  whenever 6 and <>j appear.  A5.3  Continuation of the Uniformly V a l i d Solutions Tr-»  +• It ^ i- o /-N /-> +• -i  +• U, e\ i n • >1  \ t r*  the s o l u t i o n of Equations 5.78 and 5.79,  r\ •€  A <~*  P  r\ 4-r\  which w i l l  CT -t«  then lead to  results  that are uniformly v a l i d when t = 0 ( e ~ ^ ) . For Equations 5.72 and 5.73 we have .1.  O-  -  ar  L afc-af  - Z-  a  v/  5  atar as e ( t ) 7  = f (t) ?  a o.^ Jr -  —  (AS.  +  a_v/«;  ar-  = 0 (from Equations 4.101  Q  using Equation A5.75, the terms  and 4.102), and S ( t )  Q  Q  (see Equations  and <J> are constant,  fsVsl  =  ?  In Section 5.4  to 5.117) i t was shown that a , b , e  (AS . ?<))  a-c*-  ea)  = 0 from Equations A1.6 and A 1 . 7 .  71?)  1  Q  and  [^l/sT  and  T (t) y  5.114 therefore,  can also be  3k$ A5.17 eliminated from E* and F*.  The functions  [  be assembled from Equations A5.52 and A5.53  5 d_Js  atar atar  ++  <5<A^  c a n  a-c a-c J  j"  -  j  and  in the form given below.  - <Hs a>ar  =  u> $ a_a. L <?v  -  s  ^  I  jc^Co b+9"> +- a£i9 cos 6os b+• s  a*-?  av  9) 7 5  S ( AS.  •  aV*  ix^  + d** av  =  fi u  f  3 t cpsfu^b +<£*) - b^>*s<~fo b u  I a r  Ja. cos (^1 ar  + 9)  a r  -  -  -<CjoJ a«. s-*w(^t+-9*) L at-  r  \  a 3 9 5i«- ( o j t < - P ) ar +- cLaD^cosdo^b+e*) av (AS.  a t cesCu> u 4-(jJ>) — t  ar  t>a^> s-^Cuijb-tcf>^)  dxr  L ar  a r  As before i n Section 5.4 ( o n pages 5.27  j  to 5 . 2 9 ) , any r e s u l t s  i n v o l v i n g long-period terms can r e a d i l y be obtained by inspection from corresponding s o l u t i o n s f o r the s h o r t - p e r i o d terms.  If,  therefore,  s-i)  8o)  34-6 A5.18 we neglect expressions containing long-period q u a n t i t i e s , Equations A5.78 and A5.79 can be w r i t t e n -  £*(.  =  k,"0  Sin,od t &  '  Z«>< , c*,$ 'O ^-o —- n ae, <a 0  av  bx  3V (AS.ffZ)  2Q  5i  ?i^9 40  <i c5£i c o ^ S c ' tf  + cos <-> t s  where Equations A5.80 and A5.81 have been evaluated using the expansions given i n Section A 5 . 2 .  Note, i n c i d e n t a l l y ,  the correspondence between  these expressions and equivalent terms i n Equations 5.92 and 5.93.  To  pursue t h i s comparison f u r t h e r , we can redefine the q u a n t i t i e s Y g , Y-|» 6  n  and 6, i n Equations 5.94 to 5.97 by s u b s t i t u t i n g  place of i2o and a ^° 0  5.101  and  0L d9, o  In t h i s case Equations 5.98  in to  again define conditions f o r the e l i m i n a t i o n of secular terms, but  from u ( t ) 7  0Q are  respectively.  ^3'  and v^(t)  instead of u^(t) and v ( t ) . g  constant:  2*., s^P  0  4-  a. 29, 0  O  c  A l s o , because a  Q  and  3if7 A5.19 So., fco &  0  av  - fl.  0  •  0  2. r ^ . ^ & o  -  a r  a r  aa, c ^ P /  -  a a_9 o  (  ^  -  S^-^o j  L  J  i f * , *  ar  ar  av  av  a_£. Si^9  +  a ^ c o D ,  (AS.tffc)  L  ar L  J  and we now obtain conditions analogous to those defined by Equations 5.102  a.  to 5.105 of the form: s-^f.^o^o^-OjV^oJ  +-  £>,*[z^C^-O^-4<%^~\  ar"  . s  5  a.  o  (ft s.ss)  O  (fr*.**?)  3^8  A5.20  1  ar  (A*.«»0' With the exception of  and 6^ a l l q u a n t i t i e s i n Equations  A5.88 to A5.91 are constant, so that we may again use the compact notation  I  [ •4.  a? L 3_  "  (AS.«S)  C  4*.  ~  C  J % 8 ,  ( c f . Section 5.4, page 5.29), where c-j, c , c 2  3  and c  4  are constant.  however, that the s p e c i f i c values of these constant terms i n Section  Note, 5.4  31+9 A5.21 d i f f e r from those f o r the present case. We s h a l l , in f a c t , f i n d that the general  form of Equations A5.92  to A5.95 i s preserved when higher-order s o l u t i o n s are considered pages A5.33,  A5.52 and A5.67).  (see  A f t e r s u b s t i t u t i n g f o r Q from Equation s  A4.32, the constant terms can be w r i t t e n : c,  =  ZcJ  s  s^9  D  |O ---o ) s  3  +  -4- a-J*-  7  (K.U)  From Equations A5.92 and A5.95, and using the symmetry property to derive corresponding r e s u l t s f o r b-j and <j>-j, we obtain 3a.,  d_0,  3b, at-  av  •=.  O  •= O  O  (A*,  IOO)  (AS.IOI)  (AS.iox)  350 A5.22 The non-homogeneous terms i n Equations A5.80 and A5.81 are  therefore  both z e r o , and consequently  U.^. Cb)  =  O  (As.ioO  ^  4.  \J Ct) = O '  .  f  .  t>s. ie>S)  We now proceed with the s o l u t i o n of Equations 5.74 and 5.75 Ug and Vg.  for  The a n a l y s i s , beyond t h i s p o i n t , i s rather more i n t e r e s t i n g  because here the functions u^(t) and v^(t) re-enter the system of p a r t i a l d i f f e r e n t i a l equations. From Equations 4.101, 4.102, A l . 6 and A1.7 we have:  fxY*-  ^-fO^"  *"  1  t,  cos t  Note that  [  + IS cos 3 b ^  u.(t)  2  = u-(t).  1  v.(t) =  I V i  t I  l i  +  _!l£ii!_  16  Si-v3b^  CH-  p,>  +  7 sc«.t  - j 5  J  32-  3  -  lu  N H  f o r a l l i > 5 (see the d i s c u s s i o n on pages  5.16  and  J  tf cos 3 b ^ l  \\W  (t)  CAS  s£«-3b  5.17 )  ."ot.)  351 A5.23  tosb -  Z$  (AS.  cosZb  64-  At present (from Equations 5.114  8 a  3T  = 0(e ),  f£=0(e ),  2  neither  2  or  dVs dx x  | | = 0(e ) 2  to 5.117 and A5.100 to A5.103)  and  aVj- contribute at 0(e) at*  (see Equation A5.54). -2 f + ^ | L btbr jOLeJ  r x  The functions ,  |^=0(e ). 2  Consequently  to Equations 5.74 and 5.75  -2|^s  - }Js  \_dbdv  |  and  bv J oce ) z  which can be evaluated from Equations A5.80 and _.  and A5.81, take the f o l l o w i n g form:  (AS.log)  -2  Son. (O t s  ar  I m; I  atr  3  ar  3  aV  J  i  a-r  av  l <*>S> - Q., aj> a-r air at  z  0  t  <-av  -  to***  j  lo?)  352 A5.24 atar  ar  J  e  1  ar  j  + co5 «0 t s  ar  a r  a r  I <?r  + Cos C0 t u  z ^ ^ J  5 ^ ^ *  at  t  (  at*  +  cos^ *2 o  1*. ar  and  and  - z f ab,.  J  a r  a SO. a*c 0  a r  a r  L 3 IT  J  b,^  a r  Note that i f replaced by  L  L  a v  «*>f  o  ~  b  a r  o  ^  a r  J  i n Equations A5.82 and A5.83 are  a Z9i. r e s p e c t i v e l y , the r e s u l t i n g expressions <>T 0  are i d e n t i c a l to the corresponding s h o r t - p e r i o d components of Equations A5.108 and A5.109 above.  We s h a l l soon have occasion to e x p l o i t t h i s  symmetry (see pages A5.33 and A5.34). The two remaining terms i n Equations 5.74 and 5 . 7 5 ,  a*g  - av  at-  bt  c  and  s  0(1)  20+- p , )  1  a Vg + au.^ include at d t J o(i) 7  both homogeneous and non-homogeneous s o l u t i o n s f o r u ( t , x ) and v ( t , x ) 5  5  so that UjCb.r)  «  a-tt&Cus  t +  + •  b Si*-Cu>Lk+0")  -+-  U  5  ( t )  N  r  t  (AS.Uo)  353 A5.25 I t i s convenient to consider homogeneous and non-homogeneous components of Equations A5.110 and A5.111 separately.  For the non-  homogeneous terms, we then obtain (from Equations 5.84 and 5 . 8 5 ) :  (AS.  ^  1+ p,  112.)  )i  C^  5  -^X^^  l  :  (A5."3)  L  «O .5 u  (^-u^xv-o  20^s^cO b + <->,_0,.c<ocO,_b ? u  +  /  2 o sc^.b + O Cob 5  x  351+ AS.  " ^  r  26  ~i  3 H«  (AS. U4)  (^-•vx^-o -5 f .  j  . N  4-  " r  [  J  . A, r  ZOj.tj^b -^3+o )sCv k (  "\  1 .  . .s  V 1  (AS."5 >  S (1+ p , ) ^  E  C s ^ - ^ X  « V - 0  These equations l e a d , a f t e r some rearrangement, to the f o l l o w i n g expressions:  355 A5...27  { Cos u>  s  4-  ^  .  f -/>. . . . . . . 1  Jt 7  Jb  . y 2.1  ~~?  - £  7  b  356 A5.28  4-  I)  Note that the usual symmetry between short and long-period i s preserved here ( c f .  terms  page 5.23).  For the homogeneous components of Ug(t,x) ana V g ( t , x ) we have:  Ti  St*-  ««? a . Cso(jo> b + 9) v s  +  *b  s  bcoCio b+-c^>) u  -  <C w  -  <Q ^ bs^*- C t o b + <£>*)  S  U  s  a . s c * _ ( c o b •+•  U  5  u  0*0  ( AS.il<?)  Xt  x  + UJ D **•«.(* «o b +j>} U  u  +  ffl  L  «*C b c ^ ^ b  -t-d>*)  (AS. n«j>  357 A5.29  In contrast to the non-homogeneous terms given i n Equations A5.116 and A5.117, these homogeneous functions enter i n t o the a n a l y s i s f o r higher orders  of e (see Equations 5.76 to 5.79 i n Chapter 5 ) .  It w i l l  con-  sequently be expedient to evaluate equations A5.118 and A5.118 to w i t h i n 3 an e r r o r 0(e ) , which i s the accuracy required by Equations 5.75 and 5.79. The process o f expansion i s s t r a i g h t f o r w a r d , and from Equations  5.62,  5.63, A5.57 and A5.58 we obtain the expressions given below. < r  -  "J'-bo  j  —  ^O tc^j> u  o  b, £ t w ^ t c o ^  (AS.ixo)  <^.S  - C u , b vL^ ^. 0  - sc~«Jfc s ^ ^ c . ^ u  +• ^ u ° u b ^ s^«j b 0  +  u  +• «*> o b t  *c^ * ^  ^ J " b <^> ^ w o ^ b s o ^ +• co^^b 0  (  0  * ^  0  ^  358 A5.30 z  + Q s s  |ftx-  -  o-cf^j  t ^j  -[ --- " ^ J a  a  $^Co t+•  u  ^JQ.e.-t-a e J j ^ c ^ u e j c  + Q s tosja,©, " " ^ ^ T  3  a>sCtJ t  0  c o s ^ t + e,,) +  +^  A  9  i  ^ C ^ k  +  e*)  t> ^j s^(o^b4.^)  +  0  (f>f. L ^  at  J  H  HI)  .  fcsc-9:  - o  5  a ,  ^vi^fcc«P +- < i o e  S  b C P  0  ^  - ^  s  a  0  © , ^  359 A5.31  Note again the symmetry between short and l o n g - p e r i o d terms i n Equations A5.120 and A5.121.  This symmetry i s e x h i b i t e d by a l l non-  homogeneous expressions i n Equations 5.74 and 5.75 that involve u) and s  0^, and we may therefore consider only s h o r t - p e r i o d terms when accumulating E*(t,x)  and F * ( t , x )  from Equations A5.108, A5.109, A5.117, A5.120  and A5.121.  Contributions to E * ( t , x ) and F * ( t , x )  from Equations A5.106  and A5.107 may a l s o be ignored, as the functions  {  S  c 8  -0  -  2k,^^^ ct)j g  and  I T a) ?  no secular terms i n the s o l u t i o n s f o r U g ( t ) s i m p l i f i c a t i o n s we can w r i t e :  NH  Zk-,^e Ct)J  generate  g  and ^ ( t ^ H '  W  ^  t h  these  360  A5.32  I ar  +  + Ceo  2e'  s  at?  j  (_av  av  Cn-jOHi-O  l <)v - - -V  av  .5  (^/-"VO  cat?  i  atr  ~~~  (fl5. 127.)  f  1  - ^ ^ ! ^  + -2.6 + ^,)"«-1 cOg  6^* - co a. fe P ? s  0  0  361 A5.33  + C05tO t $  ] ox  •15  bV  J>  k,^ / * , ^ , 0 - / * / { ^ [ 4 4 - o , - o ^ f - 2 o 7 s  3  \ bx  ar  J  -ICU^HQ,^  I f secular terms are to be eliminated from Ug(t)  and Vg(t)  Equations 5 . 9 8 to 5.101 must be s a t i s f i e d , where Y Q , Y-| » <5Q and 6-j denote expressions  i n Equations A 5 . 1 2 2 and A 5 . 1 2 3 when we take  A f t e r some rearrangement,  Equations 5 . 9 8 to 5.101 lead to the  following conditions:  - ie^O+^.^Ci- k ) | ^ ( ^ - O e  +  0 ^  362 A5.34  (ft*.  ^j/^Ci-^o"  -if W,/^  av  1  fc» 0^«toOo 5  -  ( c J j - * . 0,) 1  s^P  0  1  V [  363 A5.35  where c-j,  and  are defined on pages A5.20 and A5.21.  d e r i v i n g these expressions,  In  some reduction has resulted, from the  equalities  (AS.1*1)  which correspond to Equations A4.7 and A4.8 when a ) ' and i o s a t i s f y L  Equations A4.28 and A4.29. A f t e r s u b s t i t u t i n g f o r the constants c^(i = 1,2,•••4) from Equations A5.96 to A5.99, the conditions defined by Equations A5.124 to A5.127 above can be w r i t t e n as: (A*.i3cO  2  u  5  ( <v -  ar  4.  0  *i«-8  0  39* ?  (ftS.l*l)  CAS. I 3 2 - )  36ii A5.36  av  j  where the constant expressions d . ( i = 1,2,•••4) can be arranged i n the form given below:  «o.  /L.  *- '  '  ( A S . 134)  (AS.J340  365 A5.37  -  In d e r i v i n g these values from Equations substantial  reduction has resulted from using Equations  and 5.107 f o r Q , sin0Q* and cos9g* s  Now because Equations u e ttir'ni i l i e u  to  A5.124  quan oi uie b ,  o^x.  auu  should be redundant.  A4.32,  a  5.106  respectively.  A5.130  bX  A5.127,  " ^ t  to  A5.133  ,  involve only two un~  uwu u i  uiesc  iutti  c<-|uu o i  ZX  We can rewrite Equations i  .,  A5.132 •  A5.133  as:  i  =  .f.  and  3L.  (flS.I3S)  Ctn>9  6  (AS.  131)  366 A5.38 and therefore:  (AS.Kl)  The expressions C ^ i - 2 o i ) and (2«J» i + v ^ ) 3  s  v  -4e~ 6+j0*(l.-0 5  s  3  can be arranged i n the form:  C^-  A l s o , from Equations A5.128 and A5.129:  0,)  fAS.lA-3)  367 A5.39 and we therefore  -  have:  CO,  By comparing these expressions with Equations A5.134 and A5.135, i t i s evident that Equations A5.132 and A5.133 are equivalent to Equations A5.130 and A5.131. If we now w r i t e Equations A5.130 and A5.131  as:  368 A5.40  then from Equations'A5.147 and A5.148 i t follows that  -i5k 6  S.  With the exception (at present) of ^ and a 3 P , a l l terms i n a*r at these expressions are constant, so that 0  Z  ISO  369 A5.41 ^•2.  constant  -  and a l s o , i f a ( x ) 2  daz.  =  (fl?. i f i )  ,  i s to s a t i s f y the c o n d i t i o n of uniform v a l i d i t y i n T:  (As. i s i )  O.  S i m i l a r l y , using the property of symmetry, we can w r i t e =  constant  OKi?3>  bx  oxr Eauation A5.152 Drovides. toaether with Eauation A5.149, the necessary condition to determine 8Q.  In p a r t i c u l a r , i f ( l - k g ) = 0, we obtain  the f o l l o w i n g r e s u l t :  and .by symmetry:  Corresponding values f o r 9Q and <J> can obviously be obtained when k 0  Q  }  1,  but not of such an elegant form. At t h i s point i t i s tempting to assume that  2[£ = 0 (see t  dx  A5.248), and from Equation A5.150 we then have  Equation  370 A5.42  The s h o r t - p e r i o d components of u ( t , x ) 5  and v ( t , - r )  H  5  H  a r e , from Equations  5.58 and 5.59:  CAS. is*)  where 8„ can take two d i s t i n c t values i n the ranqe -TT <• 6  n  A f t e r expanding cos(w t + 6 ) and Q c o s ( w t + 0 * ) , s  Q  s  s  Q  < IT. from  Equations A4.32, 5.106 and 5.107 we can w r i t e . (ASM 6°)  <C ftoCco b $  s  Co o? k  4-  ^J^CeoPo - Z O j S ^ P o ^  r  3  - S^. t J k s  («S -0 )L l  3  Z ( 0^O  %  4- cO 0 ) x  X  =  9  0  9  0  4 - 4 - S j J " &  0  (As,  371 A5.43 Now f o r e i t h e r value o f 9g:  P  r  = 'if  (AS.lt*)  0*  COS  i n which case  cosCw t + 6 > 0 ) { < ° A 5  -•^s C  c o 5 9  o - 2 0 sin.e j.  o , - ^ * ] ceocojfc  -  = t^O^toOjt ^ Z o  e  3  2-O sc^to t  $  ^ t -  (AS. IAS)  (A*, u t )  .  5  t  5  I f we s e t kg = 1, then from Equations 5.84, 5 . 8 5 , A5.157, A5.165 and A5.166 the complete s o l u t i o n s f o r u^(t) and v ^ ( t ) can be reduced to the f o l l o w i n g form:  U. Cb)= $  -l$lf,^  u  5 20 I  v  $  Ct) =  ;L0  Kv^t  +  Oncost £  ( ^ - 1 X 0 ^ - 1V -)  lL? »-  lSk  3  Slwtr  * (-Z + Oj)  ^,|* (l-/A».) x  +•  X  (AS.It,^)  sO+fO"*\  cost 7  ^,^ Cl-/jQ  a  x  + OU)  (AMfcg)  372 A5.44 where the symmetry property has been invoked to e l i m i n a t e long-period 5 terms.  Note t h a t , although they do not appear i n the 0 ( e ) s o l u t i o n , a ^ , b  0Q and <J)Q w i l l be present i n the s o l u t i o n s f o r higher orders o f e. The present a n a l y s i s i s s u f f i c i e n t to determine e u ( t ) and e v ( t )  o in Equations 5.32 and 5.33 to w i t h i n a constant e r r o r 0(e ) , although the s o l u t i o n s are uniformly v a l i d f o r t = 0 ( e " ^ ) as e ->- 0. quently nothing to be gained by evaluating u ( t ) ^ i  H  There i s conse-  and v . ( t )  N H  .for  i > 7.  Before we continue with the s o l u t i o n of Equations 5.76 and 5.77 f o r Ug and Vg, the expansions o f d e r i v a t i v e terms must be extended to include higher orders i n e. =  da.  £  3  ^  +-  3  b~F  A t t h i s stage of the a n a l y s i s :  e^}a.  4-  H  OCe )  (hS. Ifcq)  S  bV  b&  +  bX~  ?9 bx 1  and therefore  3a. coiC u bi-9) s  X  0  Zx*-  from Section 5 . 2 :  bv  (AS. 173)  ««.(«,t+9 ) 0  Ib-x  4-  B 3a  3  Gos(eJjt+  9 )~l +OU ) s  0  373 A5.45  a.  d&i  0  c«o  £ <->, b -t-  So')  av  iX  +  L  I  ?  £<„  Ceo  (o  }  atr  ar j  a*c  dv  av  ax  j  L  dv  av j  (AS.  174)  av  L  + .e-  ar  I  av  a^j  a-r  ar  ^r)  6  at-  at- 3 CAS.17S)  ar  a 0.5 a^• ,  a^  - e  ar-  x.  a\-j c o ( i o k + © ) - d ajj&j si«.r"u>,b+ © ) >  J  ax 1  0  0  e  at"1  a a2fs 0  *M.(*«J,fc*0,  at*CAS.I7?)  37k A5.46 Note t h a t , i n the i n t e r e s t of b r e v i t y , l o n g - p e r i o d terms have been omitted from Equations A5.172 to A5.177, as complete expansions i n c l u d i n g the long-period components can r e a d i l y be obtained by i n s p e c t i o n from these r e s u l t s (see Section A 5 . 2 ) . The functions  3_j*  av  s  and  \js to  consequently do not c o n t r i b u t e  1  z  o  to Equations 5.76 and 5.77 at 0(e ), and f o r the non-homogeneous terms we now o b t a i n :  CAS  For the remainder of t h i s a n a l y s i s s p e c i f i c non-homogeneous s o l u t i o n s are not required (see the d i s c u s s i o n on page  A5.44).  If  secular terms are to be eliminated from Ug and Vg, we may therefore r e s t r i c t our a t t e n t i o n to q u a n t i t i e s i n E* and F* that i n v o l v e s h o r t period terms alone, since corresponding l o n g - p e r i o d r e s u l t s are r e a d i l y obtained from the symmetry property.  With these s i m p l i f i c a t i o n s 5 ( t ) ^  and V g ( t ) can be w r i t t e n i n t h e f o l l o w i n g form ( c f . NH N H  5.33).  u  pages 5.31  to  H  375 A5.47  lL,(t) =  - JA  x  Sin- 6*) t s  -  0.  (AS.180)  r o ( o / - - u j-y s  -  | t , <vo  4 j  s  b  J O  f i ^  - ^ k . ^ C - z O r 11-2.(^-0^1  - ^ [ i f ^ - o , ]  and i f Equations A5.180 and A5.181 are r e - s t a t e d using the compact notation  («V--<ojO  376 A5.48  then we can w r i t e :  (RS.IS4) if-  5V  *b  +3a  t  » S^0 bC^-> ')  b  s  5  5  H- c « O b s  -  X, )  . (fls.iss)  From Equations A5.52 and A5.53, together with Equations A5.172 to A5.175, and again considering only s h o r t - p e r i o d terms:  CAS. 1 8 b )  L 5.^0 b s  1o  s  <to&.  15  377 A5.49  - Z.  (A?.  -  S^0o  av  i f  >*cr  \ 4  C<o  tOjk  z<&  5  - z.  o 1 5  r  J «te e  r  We are now i n a p o s i t i o n to evaluate Equations A5.178 and A5.179, as the remaining q u a n t i t i e s  2-|  fV  5  - J ^vj y, —  c  and  have already been determined (see Equations A5.120 and A5.121).  After  some rearrangement E * ( t , x ) and F * ( t , x ) can be w r i t t e n i n the form given below.  187)  378 A5.50  ar  L>  / zci+^y*- } i<i L <D  + <*>P  0  <j  E, e  i  + *<ps  L  I  2a, ar  L  1  >\ i r  r  1  ar.  - *  0  M A a-cj  a r  L ^r  J j  J  379 A5.51  Sin. u>,b  + 1  + z^Otn.y  1  s  +- COS u> t s  QQ-i — a.  bf  £ x^ - x *> 3  ar  ar  -«r>9.  (_  I  bV  dV  ar  ar  J  J  380 A5.52 I f these equations are w r i t t e n as  ( A ? . I<»l >  then we can again use the conditions given i n Equations 5.98 to 5.101 e l i m i n a t e secular terms from Ug and Vg.  A f t e r some rearrangement  f o l l o w i n g r e s u l t s are obtained:  ^ 3  av  - ^ R ^  iv  JV  ^ 3 -*o°*>j>z IX bv  CO  - c,  at  av  .  iV  the  to  381 A5.53  ,r  +  4  ^ 3  -  0>* 1  a D, 0  ar  av  )  a P.  ar  i  c  3  Q|  a&x 4-  ar -  382  A5.54  -2U (I + JI,)* S  where the constants c^ to c^ are defined on pages A5.20 and A5.21 .  These  four equations correspond to Equations A5.124 to A5.127, and by d i r e c t analogy with Equations A5.130 to A5.133 we can therefore w r i t e  l at-  5  I  atr  3] ( ^,  -|*x,3£  I  L  a_p  -  t  av  at-  t  7  +ft a^ ] [ o  r  ?^p  4- z ^  e  Z«a,<.P  e  -  s  0  t  0  3  av j  0  ar  1  0 coP ]  S£^.9 + X ^ Ceo 0  L  0  (ft 5.1*7)  ar S  o  1 (flS'.m).  383  A5.55 where the q u a n t i t i e s  ( i = 1,2,•••4) denote constant terms i n Equations  A5.192 to. A5.195. I t should again be possible to reduce Equations A5.198 and A5.199 to the form of Equations A5.196 and A5.197.  Using the previous r e s u l t s  of Equations A5.145 and A5.146, to demonstrate t h i s reduction we must therefore show that  (4*4*+  V )  The evaluation of Equations A5.200 and A5.201 presents l i t t l e d i f f i c u l t y and, although the d e t a i l e d c a l c u l a t i o n i s omitted here, both of these conditions a r e , i n f a c t , s a t i s f i e d . Before s o l v i n g Equations A5.196 and A5.197 f o r a  1  and 6 - | ,  it  would be helpful to reduce the functions H and H to a more compact form. 1  2  A f t e r s u b s t i t u t i n g f o r Q , s i n 8 * and cos 9 * (see Equations A4.32, s  and 5.107  Q  Q  5.106  r e s p e c t i v e l y ) . t h e c o e f f i c i e n t s of a 8^ and a^ i n Equation A5.192 Q  can be expressed as:  (RS.20Z)  381; A5.56  ( A S . 203)  and f o r Equations A5.196 and A5.197 we now have:  L air  6Xr j  L  a r  j  1  4J  S  C u J ^ - cjj")  (AS.zot*-)  1 av  a* j  t  av  av j  ] (ftS.ZoS)  Equations A5.128 and A5.129 are p a r t i c u l a r l y important during t h i s of r e d u c t i o n .  process  385 A5.57 The expressions i n v o l v i n g  ( i = 1,2,•••4) may be re-grouped,  and Equations A5.204 and A5.205 then lead to the f o l l o w i n g c o n d i t i o n s :  ^3  bX  dX  ar  ue^x.E^-Oj-z]  (05.26?)  Now from Equation A5.151 and 3 9 ar  3  must be constant.  L  i w  5  c to^ -<JJ"!) 1  ^_S>z = constant,  so that both da.  3  at  In t h i s case we can w r i t e :  ^  r" L  (fl?.20S)  386  A5.58  and, i f a~(-r) i s to s a t i s f y the c o n d i t i o n of uniform v a l i d i t y i n T:  Sen. <9  e  £  ^ +  X ^Oj  1  [  W  s  l  -  0  -Z  3  1  ( A S : io<0  Without d e f i n i t e values f o r  3_e  a  and  ot  ao at  3  Equations A5.208  and A5.209 cannot be solved f o r a-j and B - j , but i f we assume that both 02 and 6^ are  constant:  ' t f JivPo^ 5  [ ^ - O j - Z ] + .....  CflS.Zio)  ^  0  ,<VoPo^ X, [ ^~O  i  - 2.]  - •  ^  387 A5.59 where a  Q  i s defined by Equation A5.157.  Note t h a t , corresponding to the  two possible values of 0Q w i t h i n the i n t e r v a l -IT < 6Q ^ IT, there are two values of 6^. The functions i n v o l v i n g X . ( i = 1,2,•••4) do not reduce s i g n i f i c a n t l y , and i t i s easier to r e t a i n the present form of Equations A5.208 and A5.209.  S p e c i f i c values of X^ can be obtained from Equations A5.180  to A5.183, but note that these equations r e l a t e only to s h o r t - p e r i o d terms. D i f f e r e n t values of X^ consequently apply i n the l o n g - p e r i o d case, and t h i s must be taken into consideration when e v a l u a t i n g A5.209 with b , <f> and  Equations  A5.208 and  i n place of a , e and OJ .  F i n a l l y we consider the s o l u t i o n of Equations 5.78 and 5.79 f o r U-|Q and V.|Q.  Only those q u a n t i t i e s which generate secular terms i n the  s o l u t i o n need be considered,  . A <t> D  - 2k,|^$  l  o  so that the functions  a>  and  T  l D  Cb) - Z k . ^ c . / t )  may both be ignored (see Equations A l . 6 , A 1 . 7 , 4.101 and 4.102). Equations A5.100 and A5.151 both a, and d j i are constant. from Equations A5.197 and A5.198,  3f-3 and a_9  3  1  This s i m p l i f i c a t i o n i s discussed on page A5.46.  From  Consequently,  are constant.  case Equations A5.176 and A5.177 reduce to the f o l l o w i n g form:  av -  '  In t h i s  388 A5.60 and therefore neither of these terms contribute at 0(e ) to Equations and 5.79."  At t h i s stage of the a n a l y s i s a ~  and | |  are o(e ) 2  and  0(e ) r e s p e c t i v e l y (see Equations A5.151 and A5.152), and from page Uy(t) and Vy(t) are both zero.  5.78  A5.22  Consequently the d e r i v a t i v e s of u^ and  v , together with the expressions 7  c o n t r i b u t i o n to the non-homogeneous terms of Equations 5.78 and 5.79  (see  Equations A5.52 and A5.53)1 I f we include only short period terms,^ A5.52, A5.53.and A5.172 to A5.175:  . bxr  t  c f . page  A5.46.  A  then from Equations  389 A5.61  I  2o<  ]  I • z.7  ajv  s  I at-ar  + a^. 7  CflS.2.1S")  a r 3  2<3?, £0  5  { ^  Co9  dr  -s^D *[ 0  a  L  i  .a3  ar  LP* "1 ar \  ar  +  a r  + «. f aj^. - & * a & A e  v ^  d i r >  ;  +^3*3! ^ J  390  A5.62  Note the equivalence i n s t r u c t u r e between these two expressions and the corresponding r e s u l t s f o r lower orders of e (see Equations A 5 . 8 2 , A5.83, A5.108, A5.109, A5.186 and A5.187). . We next require expansions o f : t  and at t h i s point the advantages of assuming k  n  u  siderable.  = 1 and  0  are con-  bX  With t h i s s i m p l i f i c a t i o n , from Equations 4.101, 4.102, A5.167  and A5.168 i t f o l l o w s that none of these q u a n t i t i e s involves short (or long) period terms, and we may therefore s i m p l i f y the non-homogeneous expressions i n Equations 5.78 and 5.79 e x t e n s i v e l y , as shown below.  d <X  - z  S  av  i  at  x  at  S  oaO  (PL*.211,) Note that u  5  and v u v  cf.  5  5  page . A5.24.  g  here denote complete s o l u t i o n s , so that  = u (t)j + u (t) 5  =  v  5  ( t )  |  H  5  +  v  5  ( t )  N H  NH  '  391 A5.63  b\  +  s  du.  fO-f a\/ -t- Ji* ^ L  s  }tr  atar  5  We')  These functions can be evaluated d i r e c t l y , using Equations A5.120, A5.121, A5.214 and A5.215.  For E * ( t , x ) and F * ( t , x ) we then o b t a i n :  - z  |_  a r  yc  L *v  -  3- a r /  ar  ar  a rJ  J 3  j  it  Z arc  V ar  ar  (AS". 218)  av  ar  392  A5.64  r  av  + z  -  <?ir  avj  fex  h  3  ]1 $  *  }  }  ]  A f t e r s u b s t i t u t i n g f o r Q , s i n 6Q* and cos 8 * from Equations G  A4.32, 5.106  and 5.107:  Q  393 A5.65  © 5 u> CJO 5  +5^9,,  =  u>^0 cos9 - s;^9 x  b  0  (u) ' 5  l +  o ^  ,  3  so that Equations A5.218 and A5.219 can be w r i t t e n as:  1  CAf.^2-3)  39i+  A5.66  F * ( t , r )  4 co^jb!  I  < -P P + <te0 >C V+ 3) ? <  e 0  t  o  (AS.  12-5)  where H  L  (fl*.2Zt)  i  L  y a^ - e,  ar  1<  ^r  ^ 3 ]  arj  *  (ft&zz?)  « e,i?3 ar 0  L  M  - C*e>. +<*O ae* ] 0  I f we now w r i t e E * ( t , x ) and F * ( t , x ) i n the form  t  ar  J  395 A5.67 F * ( t , t )  =  o~  b  Sir*. £0j t  -v  J  (  COS CO t $  C AS-.zs.i')  ,  then to e l i m i n a t e secular terms from U-JQ and V-JQ Equations 5.98 to 5.101 must be s a t i s f i e d .  A f t e r some s t r a i g h t f o r w a r d manipulation the f o l l o w i n g t  conditions are obtained: CflS.2.30)  o - o (AS".i32.>  t^  5  C* tOf  t^t?}  -  (AS.233 )  By comparing these r e s u l t s with Equations A5.130 to A5.133 we can state d i r e c t l y that Equations A5.232 and A5.233 are equivalent to Equations A5.230 and A5.231 r e s p e c t i v e l y .  Note a l s o , from Equations A5.124  to A5.127 and A5.130 to A5.133, that Equations A5.230 to A5.233 can be written as:  c, U - c,, 8, x  c  z  + t , H,  = o =  O  To obtain expressions of t h i s form i t i s necessary to use Equations A5.128 and A5.129.  396 A5.68  c  H  s  i .  ~  c  4  H  ,  =  °  ( AS.2.36)  which i s consistent with previous r e s u l t s (see page  A5.20).  From Equations A5.230 and A5.231 we now have: H , C O  H  x  £ r )  =  O  =  O  CAS-.zSg)  ,  (AS.i3<0  so that  .....  <?r  . . . a_tr  a r . _ _  w j e  2-  a r ^  3. J  [ .  a r  a r  L  (AS.Z4.0)  J  (AS.2M)  At the present stage of the a n a l y s i s :  ar £§3  ar  =  constant  (from page A 5 . 5 7 ) , and from Equation A5.151  Z9  Z  =  constant  397 A5.69 From Equations A5.240 and A5.241 i t therefore f o l l o w s that  ^f-q. =  bO^  constant  = constant  (AS.a-4-i-)  .  CftS*4-3)  To maintain uniform v a l i d i t y i n x we must require  -  O  (AS. 2.4-4)  i n which case Equations A5.240 and A5.241 can be w r i t t e n as:  a r  av  bv  We can s u b s t i t u t e f o r  a  i n Equation A5.246 from Equation A5.208  Q  to o b t a i n :  CAS. 24?)  Ceo  -  «,s~.P<> ^  \  [  t O ^ - O j -2-]  +  . -  ^  398 A5.70 ^ "I and therefore, since the function j^Ci + p,) *• -.|P». ^  is constant and i  non-zero:  0 , -  constant  («i**<0  (see Equation A5.209). I t i s convenient a t t h i s p o i n t to set  and from Equation A5.247 we now have:  P^  - /*,P, ( T c ^ P 0  . .• a . « V 5  tf^s^Q  ...  .  (ftr.^i)  -COJ)"  I f i t i s assumed that both Equation A5.245:  ^ 6 3 and  19^ are z e r o , then from  at  at  2. where, i n t h i s case, 9-j i s defined by Equation A5.211. Corresponding r e s u l t s f o r b  2  and (J> can r e a d i l y be obtained 2  using the symmetry property, but note that the q u a n t i t i e s r , A and A., ( i = 1,2,•••4) i n Equations A5.249 and A5.250 must be re-evaluated i n terms of the long-period v a r i a b l e s o ^ , b and-<j> (see the d i s c u s s i o n on page  A5.59).  399 A5.71 This completes the d e t a i l e d a n a l y s i s of four-body motion near and L g , although the r e s u l t s derived here are condensed i n Chapter 5.  A6..1.  APPENDIX VI  A6.1  CHEBYSHEV POLYNOMIALS  Introduction Suppose we wish to approximate a f u n c t i o n f ( x ) over some  i n t e r v a l , f o r example interpolation.  -l^x^l,  to w i t h i n an e r r o r ±e using a polynomial  A number of i n t e r p o l a t i o n schemes may be used [ 6 2 ] , 2  but to obtain the prescribed accuracy a' Targe number of terms may be necessary,  a feature which i s p a r t i c u l a r l y s i g n i f i c a n t when .accuracy  to be maintained over the e n t i r e range of x .  In t h i s case an i n t e r -  p o l a t i o n procedure based on Chebyshev polynomials leads to the most "economical" approximation, in the sense t h a t , f o r a c e r t a i n  accuracy  The Chebyshev polynomials T ( x ) are simple trigonometric k  functions cosk6  s  IX- =  but expressed i n terms of the v a r i a b l e  cosB ,  where - I T ^ 6 ^ IT and, consequently,  -1 ^ x ^ - 1 .  I t i s frequently  * more convenient to work with the s h i f t e d Chebyshev polynomials T ^ ( x ) , f o r which T * Cx)  =  cos k9  k  with  cos 9  =-  1  is  2+01 A6. As 6 v a r i e s from 0 to i r , we  now have 0 ^ x ^ l .  mation f o r an a r b i t r a r y f u n c t i o n f ( x )  The method of a p p r o x i -  i s o u t l i n e d i n the f o l l o w i n g  section.  A6.2  Function Approximation using S h i f t e d Chebyshev Polynomials Consider the problem of approximating some f u n c t i o n f ( x ) by a  polynomial in x over the i n t e r v a l -a 4 x ^ 3.  I f s h i f t e d Chebyshev  polynomials are used, then a new v a r i a b l e y must be defined such that the i n t e r v a l -a £ x ^ 3  corresponds to 0 ^ y 4 1.  Of the many t r a n s -  formations s a t i s f y i n g t h i s c o n d i t i o n , the most obvious i s  and we now take  [62 ] 3  Z  In t h i s case f ( x ) i s transformed i n t o an even f u n c t i o n of 6 which can be expanded as CO  rK*0  =  +  j.  ...  %  lest  cos\< &  .  (06.3) --  However, since |<(,y) = coske, t h i s expression i s equivalent to T  eo  The constants a^ are obtained from a numerical i n t e r p o l a t i o n  ii02 A6.3 procedure [62 *]. and i n p r a c t i c e the summation may be terminated a f t e r 1  n terms.  I f |a | denotes the magnitude of the l a r g e s t c o e f f i c i e n t ,  the  m  summation could be truncated when, f o r example,  |a -|l < 10 n+  1 0  |a l • m  We then have  Z — i  k  si  where e i s s u f f i c i e n t l y small that i t s e f f e c t may be neglected.  To  derive the c o e f f i c i e n t s a^, f ( y ) i s evaluated at the points tj^  =  1  +  cos C S " T T / V )  S  =  O  f  \ 1 >  j  - •• , n. .  which y i e l d a l i n e a r d i s t r i b u t i o n o f data points 6 over the i n t e r v a l 6  0  is- TT.  Equation A6.5 can be rearranged i n t o the power s e r i e s  iAx}  =  N  n.  21—»  . . .. . '  t\  . ...  +  e. ,  and the corresponding s e r i e s i n x can now be w r i t t e n  ^  or, alternatively:  J  (A4-6)  k03 A6.4 where  denotes the binomial c o e f f i c i e n t  m  For  computation, however, the f o l l o w i n g form of Equation A6.8 i s more convenient:  J  A6 3  C o e f f i c i e n t s of  (y)  I f the s h i f t e d Chebyshev polynomial of order k i s w r i t t e n as k 2-.  i — O  then, from Lanczos [ 6 2 ] ,  the c o e f f i c i e n t c^ i s defined by:  ( A * - II )  W.-C -  C o e f f i c i e n t s of the f i r s t twelve s h i f t e d Chebyshev polynomials are given i n Lanczos [ 6 2 * ] .  In the present case, however, Equation  A 6 . l l provides the information needed to evaluate  Equations A6.6 and  A6.9.  A6.4  Approximate Expansions Suppose the expansion defined by Equation A6.9 i s s u f f i c i e n t l y  accurate t h a t . w i t h some f i n i t e value of n , f o r a l l p r a c t i c a l  purposes  k-ok the e r r o r e may be taken as zero.  A6.5  In t h i s instance Equation A6.9 can  be rearranged as the i d e n t i t y  which, from Equation A 6 . 6 , we can a l s o w r i t e as  £6c"> -  t  k  <j  k  .  ( f i t . 13 >  k=.o  Now from Equation A6.10: k  so that  kH k  *  where |T^(y)| ^ 1.  I f we s u b s t i t u t e f o r y  which can now be w r i t t e n i n the form  n  i n Equation A6.13:  1+05 A6.6 CA6.I7 )  where the e r r o r 6 s a t i s f i e s  1  5 1  •  -  .  CA6-I8)  This process of contraction can be continued so that f ( x )  i s approximated  by p r o g r e s s i v e l y fewer terms, but with an i n c r e a s i n g e r r o r [ 6 2 ] 7  We next i n v e s t i g a t e the behaviour of |S| when n-1 terms are r e t a i n e d but the i n t e r v a l - a ^ x ^ 3  i s decreased.  I n t u i t i v e l y one  would expect the e r r o r to d i m i n i s h , and i n f a c t the reduction i s associated with b . n Let DC  =  =c  0  (M,.  •+ CT£  |q )  where a t l , the i n t e r v a l of i n t e r e s t i s defined by  -oo. ^  and f ( x )  5  i s bounded on t h i s i n t e r v a l .  value f ( X g ) .  As a.-»• 0, f ( x ) w i l l tend to some  We can therefore w r i t e  4Gt> =  SCO  +  v$*C$)  ,  Cfl6.ao)  it •  where f (£) i s bounded and f ( x ) - f ( x ) n  = 0(a) as a -*• 0.  The i n t e r v a l  A6.7 -a ^ C - B  corresponds  to  and s u b s t i t u t i n g f o r x i n Equation A6.1 we obtain  ^  -  % + oc  The function cj>(6) i s  )  now defined by  which we can w r i t e as  Equation A6.13 i s equivalent to rt.  (see Equation A 6 . 5 ) , where the F o u r i e r c o e f f i c i e n t s numerical i n t e r p o l a t i o n process.  a  k  are derived from a  Note t h a t , because n i s the  exponent of y i n Equation A6.13, b  n  using i n t e r p o l a t i o n , the c o e f f i c i e n t s  i s equal to a  n <  largest  If, instead of  a^ were obtained from the i n t e g r a l  IT •  «  k  =  i.  (f>C0) co lt& d9  |  S  o  ,  (fl6.a-5;  .  4  0  •  7  A6.8 the two values of a^ would be i d e n t i c a l only when f ( x ) contained no harmonics higher than cosnG, i n which case Equation A6.20 would be exact [62 ]-  Although t h i s c o n d i t i o n may not be s a t i s f i e d , we assume here  8  that Equation A6.25 can be used to define a  n  and b  n >  Consequently we  can w r i t e  If  (9)  cosr\9  <19 .  ,  .  and t h e r e f o r e , from Equation A6.22 II  H  Cos  T?  J  ir  For b we then obtain n ir  Cos m. 9 <&9  so that b  n  = 0(a) as a 0 .  (fl&-2s)  When n »  1, the q u a n t i t y c  n  i n Equation  A6.18 w i l l be a large i n t e g e r , ^ and i t f o l l o w s that O ( r )  =  \l\  as  0-  _>  o  The e r r o r 6 associated with a Chebyshev i n t e r p o l a t i o n of f ( x ) on the i n t e r v a l X Q - oa ^ x  From Equation A 6 . l l : c  n  c  X Q + a3  therefore diminishes as a -> 0.  408  A6. Note t h a t , f o r t h i s s p e c i f i c i n t e r p o l a t i o n process,  if  g(x)  denotes the approximating polynomial: I f ( x ) - g(x) so that f ( x )  | ^ |5|  f o r a l l x such that x  Q  - aa ± x ^ X Q + a3,  i s approximated uniformly over the e n t i r e i n t e r v a l and,  f o r a given order of g ( x ) , with increasing accuracy as 0 tends to zero.  A6.5  Expansion of the Function  | r . . |"  From Chapter 5, Equation 5.7,  the expansion of  J  i s given by:  J_  r1  +  and i f we take  ... -x.. . =  then ..[  For L . and Lj-  we have  | r..  1 +  | *= 1 (Equation 3 . 4 5 ) , and i f f ( x ) =  _L  about  k-09 A6.10 =  [ I +  (A6.3l)  J  2 where  x = | r^. \ - 1.  If f ( x )  i s approximated i n the i n t e r v a l  - a ^ x ^ 3, as shown in Figure A 6 - 1 , then:  Figure A6-1  f ( x ) = [1 + x ]  •3/z  The approximation w i l l consequently be v a l i d w i t h i n a region about p . , bounded by inner and outer r a d i i  (l-a) ^ 1  and  (1-3)  fx  respectively.  For the a n a l y s i s of Appendix V, Section A 5 . 1 , i t i s necessary to evaluate the functions  and  over i d e n t i c a l  i n the sense that a and 3 must be equal f o r both expansions.  intervals  I f t h i s were  410  A6.n not so, the c o e f f i c i e n t s and A5.16 would d i f f e r . near p ,  0  ( i = 0, 1, 2, 3, •••• ) i n Equations A5.15 The corresponding region of the  and p ~ , w i t h i n which the expansions of  ,3  and (l+B)'  _j  r\  plane  and  are both v a l i d , i s shown i n Figure A6-2 f o r ( l - a ) ' * = 7  /a  =  1.35.  0.65  14-11 A6.12 Note that i f a and 3 are chosen so that  l - ( l - a ) ' " = (1+3) /a  the expansions are v a l i d i n s i d e a c i r c l e of radius p about For the function f(x') = [1+x]  /z  -1 = p , and L g .  , and with these p a r t i c u l a r  values of a and 3, we have: -0.5775 ^ x ^ 0.8225 , 3.6413 ^ f ( x ) ^ 0.4064 . The e r r o r e i n Equations A'6.5 to A"6.9 i s reduced below T 0 ~ ^ f o r n > 25; -8 the corresponding Chebyshev i n t e r p o l a t i o n i s accurate w i t h i n 10" the i n t e r v a l -0.5775 < x 4 0.5985.  over  A s l i g h t error becomes apparent f o r  0.5985 ^ x ^ 0.8225, reaching a maximum of 4 x 1 0 " when x = 0.8225. 7  The c o e f f i c i e n t s of x fl  y  tvi S  I E  4- A n ^ +- U  m  i n the expansion of f ( x ) are tabulated below f o r  ,.,^4-U  o^wv*^/.^««rl4^r.  U4«  -I 1  * . 1 . . ~ <-  A6.13 Chebyshev - Derived Coefficients  Binomial Coefficients  0  1.0000 000  1.0  1  -1.5000 000  -1.5  2  1.8750 000  1.875  3  -2.1874 999  -2.1875  4  2.4609 366  2.4609 375  5  -2.7070 411  -2.7070 313  6  2.9326 656  2.9326 172  7  -3.1417 974  -3.1420 898  8  3.3371 447  3.3384 705  9  -3.5284 891  -3.5239 410  10  3.7215 732  3.7001 381  11  -3.8294 740  -3.8683 262  12  3.8122 951  4.0295 064  13  -4.3456 147  -4.1844 875  14  5.7458 168  4.3339 334  15  -4.4987 347  -4.4783 979  Table A6-1  C o e f f i c i e n t s of x i n the Series Expansion of [1+x] f o r -0.5775 $ x ^ 0.8225 m  PUBLICATIONS  Barkham  ;  P.P.D. and Soudack, A.C., An e x t e n s i o n t o . t h e method o f K r y l o f f and B o g o l i u b o f f . Intv Journ. of C o n t r o l , V o l . 10 (1969), pp. 377-392.  Barkhazn, P.G.D. and Soudack, A.C., A p p r o x i m a t e s o l u t i o n s of n o n - l i n e a r , non-autonomous s e c o n d - o r d e r d i f f e r e n t i a l equations. I n t . Journal of C o n t r o l , V o l . 11 (1970), pp. 101-114.  Soudack, A.C.', and Barkham, P.G.D. , F u r t h e r r e s u l t s on "Approximate s o l u t i o n s o f n o n - l i n e a r ^ non-autonomous s e c o n d - o r d e r d i f f e r e n t i a l . equations." I n t . Journal of Control, V o l . 12 (1970), pp. 763-767. Soudack, A . C , and Barkham, P.G.D., On t h e t r a n s i e n t s o l u t i o n of t h e u n f o r c e d D u f f i n g e q u a t i o n w i t h l a r g e damping. I n t . J o u r n a l o f C o n t r o l , V o l . 13 (1971), pp. 767-769. Barkham, P.G.D. and Soudack, A . C , A p p r o x i m a t e d i f f e r e n t i a l equations w i t h s a t u r a t i n g .nonlinearities. I n t . Journal of Control, V o l . 19 (1974) pp. 941-946. Barkham, P.G.D., Modi, V . J . and Soudack, A . C , The concept of reference L o c i a p p l i e d t o f o u r - b o d y dynamics. P r e s e n t e d a t t h e 1 3 t h I n t e r n a t i o n a l Congress o f T h e o r e t i c a l and A p p l i e d M a t h e m a t i c s , Moscow, August 21-26, 1972. Barkham, P.G.D., M o d i , V . J . and Soudack, A . C , The concept o f r e f e r e n c e L o c i a p p l i e d t o f o u r - b o d y dynamics. P r e s e n t e d a t t h e 23rd I n t e r n a t i o n a l A e r o n a u t i c a l C o n g r e s s , V i e n n a , October 8-15, 1972.  

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