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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. (Eng. ) , Southampton U n i v e r s i t y , 1967 M . A . S c , Univers i ty 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1974 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for 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 ibrary shal l make i t f r e e l y ava i lab le for reference and study. I further agree that permission for extensive coyping of t h i s thesis for scholar ly purposes may be granted by the Head of my Department or by his representat ives . . I t 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 thesis for f i n a n c i a l gain shal l not be allowed without my wr i t ten permission. PETER G.D. BARKHAM Department of E l e c t r i c a l Engineering The Univers i ty of B r i t i s h Columbia, Vancouver 8, Canada Date W • 117*. i i 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 re lated form of the r e s t r i c t e d four-body problem. Although a number of constra ints 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 consistent and ra t iona l approach to the analysis of four-body systems has not previously been developed, and an attempt i s made here to describe problem evolution i n a systematic manner. In the p a r t i c u l a r three-body problem under considerat ion two masses, forming a close binary system, o r b i t a comparatively d i s tant mass. A new l i t e r a l , per iodic so lu t ion of t h i s problem i s found i n terms of a small parameter e, which i s re lated to the distance separating the binary system and the remaining mass, using the two var iab le expansion procedure. The so lut ion i s accurate w i t h i n a constant error O(e^) 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 re lates to the sun-earth-moon conf igurat ion of the so lar system. The second example applies 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 so lut ion with an equivalent numerical ly-generated o r b i t shows .close agreement, with an error below 5 percent f o r the sun-earth-moon conf igurat ion and less than 3 percent f o r the s t e l l a r system. i i i 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 into the close binary system. Unique uniformly v a l i d solut ions are found for motion near both equi la te ra 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 wi th in a constant e r ror 0(e ) , and the solut ions are uniform-l 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 ~ ^ ) . O r b i t a l pos i t ion errors near and of the earth-moon system are found to be less than 5 percent when numerically-generated per iodic solut ions are used as a standard of comparison. The approach described here should, i n general , be useful i n the analys is of non-integrable dynamic systems, p a r t i c u l a r l y when i t i s f eas ib le to decompose the problem i n t o a number of subsidiary cases. i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES . . . . . . v i i LIST OF FIGURES . v i i i ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . i x Chapter 1. INTRODUCTION . 1 . 1 1.1 Background 1.1 1.2 Methodology 1.7 2. PERIPHERAL MATERIAL 2.1 2.1 Introduction . . . . . . • • 2.1 2.2 Limitat ions and Res t r i c t ions . . . . . . . . . . 2.1 2.3 N-body Dynamics . . . . . . . . . . . . . . . . 2.8 2.4 nomographic Solutions . . . 2.14 2.5 Two-body Motion . . . . . . . . . .. . . . . . 2.16 2.6 Non- iner t ia l Systems of Reference . . . . . . . 2.24 2.7 Integrals i n Rotating Systems of Reference . . . 2.32 3. LIMIT PROCESS CONSIDERATIONS . . . . . . . . . . . . 3.1 3.1 Introduction . . . . . . . . . . . . . . . . . . 3.1 3.2 Reduction of the 3-body Problem . . . . . . 3.3 3.3 Reduction of the 4-body Problem . . . . . . 3.10 V Chapter ' Page 3.4 Small-parameter Expansions 3.13 3.5 Exact Solutions . 3.15 3.6 Orbi ta l Perturbations . . 3.19 3.6.1 Perturbed 2-body motion 3.19 3.6.2 Perturbed 3-body motion . 3.21 3.7 The Two-variable Expansion Procedure 3.26 3.7.1 Def in i t ions . 3.28 3.7.2 M u l t i p l e scale methods 3.34 4. THE THREE-BODY PROBLEM . 4.1 4.1 Introduction 4.1 4.2 Prel iminary Analysis . . . . . . . . . . . . . . 4.2 4.3 The Uniformly V a l i d Solut ion . . 4.10 4.4 A Uniformly V a l i d Solut ion of the Restr ic ted Problem . • . . . • • 4.26 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 . . . . . . 4.28 4.4.2 H i l l ' s v a r i a t i o n o r b i t . . . . . . . . . 4.34 4.4.3 Further terms i n the so lu t ion 4.38 4.5 Discussion . . 4.44 5. THE FOUR-BODY PROBLEM, 5.1 5.1 Introduction ^.1 5.2 The Restr ic ted Three-body Problem . . . . . . . 5.2 5.3 Four-body Motion Near L^ and L^ . . . . . . . . . . . 5.9 5.4 Uniformly V a l i d Solutions . . . 5.21 5.5 D i s c u s s i o n . . . . . . . . . . . . . • • . • • • 5.42 v i Chapter Page 6. SPECIFIC SOLUTIONS . . 6.1 6.1 Introduction . . . . . . . 6.1 6.2 Three-body Orbits . 6.1 6.2.1 Per iodic solut ions of y ( t ) = B(t)y + f ( t ) 6.4 6.2.2 I t e ra t ive method to determine a T-per iodic o r b i t 6.7 6.2.3 Per iodic earth o r b i t 6.9 6.2.4 Per iodic o r b i t s for s t e l l a r motion . . . 6.16 6.3 Four-body Orbits Near L 4 and Lg . . . . . . . . 6.21 6.4 Discussion . . . . 6.29 7. CONCLUSION 7.1 REFERENCES . . . . . . . . . .. .'. . ' : . . . . . . . . . • . R1 APPENDICES v i i LIST OF TABLES Table p a 9 e 3-1 Orbi ta l Motion in the n-body Problem for 2 ^ n < 4 3 - 2 6 v i i i LIST OF FIGURES Figure Page 2-1 I n i t i a l conf igurat ion for the n-body problem . . . . 2.9 2- 2 Primary configurat ion and coordinate systems . . . . 2.26 3- 1 Regions of motion for p 2 and p^ • • • 3.4 3-2 Lagrange points of the r e s t r i c t e d problem . . . . . 3.18 3- 3 Development of the four-body problem . 3.27 4- 1 Primary conf igurat ion for the r e s t r i c t e d problem . . 4.27 4-2 Primary conf igura t ion , with respect to the 5*, H* coordinate system, for the r e s t r i c t e d problem 4.30 6-1 Per iodic earth o r b i t about £ 2 =0 .012150 , n 2 = 0 - 0 • • . ' 6.12 6-2 Pos i t ion errors f o r the earth o r b i t over one period 6.14 6-3 Numerical and a n a l y t i c a l so lut ions about C2=0.012150, n 2=0.0 6.15 6-4 Per iodic o r b i t for y - ^ l O , y 9 =0.5, r,=50 about. £ 2 = 0 , 5 , n2=0.0 . . . . . . . . . . ! . . . . . . . . 6.18 6-5 Pos i t ion errors for the s t e l l a r o r b i t over one period • • • • • 6-6 Per iodic o r b i t s near L^ i n the four-body problem . . 6.25 6-7 Per iodic o r b i t s near L^ i n the four-body problem . . . . . . . . . . . . . . 6.26 6-8 Pos i t ion errors for the o r b i t near L^ . . . . . . . 6.27 6-9 Pos i t ion errors for the o r b i t near L5 • • 6.28 i x ACKNOWLEDGEMENTS I t i s a pleasure to acknowledge the advice and encourage-ment of my supervisors , Dr. V . J . Modi and Dr. A . C . Soudack, during the inception 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 for p r i n t i n g . This research was supported by Northern E l e c t r i c , the Univers i ty 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 and interest made this work possible. 1 T . l 1. INTRODUCTION 1.1 Background The n-body problem of c e l e s t i a l mechanics may be def ined, quite s imply, 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 sub-sequent motion. No general so lut ion of t h i s problem e x i s t s . Only the two-body problem i s considered solved, because properties of the t o t a l i t y of poss ible solut ions are known, and even i n t h i s case the o r b i t a l coordinates cannot be represented as e x p l i c i t closed-form functions of time. Methods which have been applied to invest igate n-body motion can broadly be divided into three categories : q u a l i t a t i v e , quant i ta t ive and f o r m a l i s t i c dynamics. Q u a l i t a t i v e methods may be employed, f o r example, to determine conditions for s t a b i l i t y , so lut ion existence and bounded motion, but are not usua l ly helpful i n es tab l i sh ing s p e c i f i c so lu t ions . T h i s , i n general , i s the domain of quant i ta t ive or exper 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 fec t of quantizat ion error over large i n t e r v a l s of time may inva l ida te any resul t s derived from a numerical in tegra t ion process. Information about system behaviour along the en t i re time axis cannot therefore be obtained from a purely quant i ta t ive approach. 1.2 P a r t i c u l a r solutions of a per iodic or asymptotic nature have, f o r t h i s reason, assumed a pos i t ion 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 ruc ture . The analysis presented here i s concerned s p e c i f i c a l l y with per iodic o r b i t s in 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 conf igurat ion w i l l be described as the s t e l l a r three-body problem, although i t may apply equal ly to g a l a c t i c or planetary systems. The four-body case i s derived from the s t e l l a r three-body configurat ion by introducing an addi t ional p a r t i c l e of n e g l i g i b l e mass into 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 so lar system; the s t e l l a r problem, for example, i s a general izat ion of the main problem of lunar theory, and the four-body model i s of considerable current in teres t i n s a t e l l i t e dynamics. The periodic o r b i t s i n question emanate from homographic, or conf igurat ion-preserving , solut ions of the two and three-body problems. A l l solut ions of the two-body problem are , i n f a c t , homo-graphic , 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 of axes i s then selected i n which the two bodies appear to be s ta t ionary . The three-body configurat ion i s a special case of Lagrange's homographic so lut ion for 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 solut ions ex i s t where the three bodies move i n coplanar o r b i t s ; the posi t ions of r e l a t i v e equi l ibr ium for an i n f i n i t e s i m a l mass i n the rota t ing coordinate system are c a l l e d Lagrange, l i b r a t i o n or equi l ibr ium points , and are usual ly denoted by L-j, L£, •••» L ^ . Three of these points (L-,, and Lg) are located on the l i n e j o i n i n g the two pr imar ies , while the remaining two form equi la te ra l t r i ang les 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 re lated 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 text •[!]. Major reviews of the general three-body and n-body problems by Lovett [2] and several other authors appeared between 1896 and 1919 [ I 1 ] . Recent work i s mentioned in a. remarkable contr ibut ion 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 eas t for an i d e a l i z e d earth-moon system, i s s table . This s t a b i l i t y i s disturbed by the grav i ta t iona 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 periodic or bounded motion i n these regions. By analogy with the three-body case, a four-body problem involv ing 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 . C e l e s t i a l mechanics entered a period of rapid development towards the close of the nineteenth century, fo l lowing the fundamental researches of H i l l and Poincare, and in teres t i n the. four-body problem can be traced to These solut ions 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 inves t iga t ion 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 solut ions of . the general problem. Twenty-eight homographic configurat ions were found a f te r formulating an integra l of motion for the i n f i n i t e s i m a l mass. Moulton's resu l t s were extended by Lovett [5] to include per iodic motion of the p a r t i c l e near these posi t ions of r e l a t i v e e q u i l i b r i u m . More recently 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 substant ia l number of numerical and a n a l y t i c a l studies were based on the v e r y - r e s t r i c t e d model. 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 resu 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 solut ions f o r a l i n e a r i z e d four -body problem near L ^ , and compared the predicted motion with numerical so lu t ions . Some a m p l i f i c a t i o n of these resu 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 [8 ] . Tapley et al. [9,10,11,12] used numerical methods to invest igate motion i n the v i c i n i t y of L^ for 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 fourth p a r t i c l e , and derived-existence condit ions for per iodic o r b i t s near Lagrange points . In [15] Wolaver examined the e f fec 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 condi t ion for bounded motion about L^ . Kolenkiewicz and Carpenter [17] used numerical perturbation techniques to f i n d per iodic and almost-per iodic 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 resu 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 fec t of so lar rad ia t ion pressure on s a t e l l i t e motion near earth-moon Lagrange points . The analys is was fur ther modified [19] to include the e f fec 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 so lar r a d i a t i o n pressure i s taken into account. Luk'yanov [21] has considered s a t e l l i t e motion near the equi la tera l t r i a n g l e points when the primary bodies form two non-interact ing binary systems, obtaining r e s u l t s s i m i l a r to those given i n [8] . The pos i t ion control of a s a t e l l i t e near a Lagrange point has been investigated by Wolaver et al. [22] , 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 for a r e a l i s t i c model of primary motion, and provided excel lent control over long periods of t ime. 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 also Danby [24] , who proposed an a l t e rna t ive approach to account for secular perturbat ions . 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 sui table descr ip t ion 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 intent ion of determining asymptotic ser ies solut ions near Lagrange points . Shi and Eckstein [26] used-.-.^matched -asym-p t o t i c expansions to determine earth-moon t r a j e c t o r i e s for a r e a l i s t i c primary model. These a n a l y t i c a l solut ions 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 analysis by Breakwell and Pr ingle [29] to invest igate 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 in some respects to a numer-i c a l so lut ion 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 dupl icated a n a l y t i c a l l y by Kamel and Breakwell [31]. Note however, that Schechter employed a d i f f e r e n t primary model, so that exact agreement between the two orb i t s should not have been expected. The a n a l y t i c a l so lut ions obtained by Luk'yanov [21] were extended i n [32] to include the i n d i r e c t e f fec t of so lar 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 for an in terac t ing primary model to predict stat ion-keeping requirements at the translunar and c i s l u n a r points L ^ a n d l^. Giacagl ia and Franca [34] 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 in to the evolut ion of orb i t s near sun-perturbed col l i n e a r equi l ibr ium points of the earth-moon system. Giacagl ia [35] has also 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 four -body problem where three primary masses are located at the ver t i ces of an equi la tera l t r i ang le according to Lagrange's c i r c u l a r homographic conf igura t ion . A r e g u l a r i z i n g transformation i s found for t h i s case, and extended to an (n+1) body problem i n which n equal masses move i n + c f . page 6.22. 7 1.7 a regular polygonal conf igurat ion . The numerical analys is of four-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 in a region about was found to be possible f o r a period exceeding 5000 days. Many of the resu l t s indicated here are discussed by Szebehely [ l 2 ] in what i s perhaps the most comprehensive survey of motion near equi l ibr ium points of the r e s t r i c t e d problem. 1.2 Methodology These diverse resu l t s do. not, unfortunately , a f ford any s i g n i f i -cant ins ight into 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 analysis takes, as i t s p r i n c i p a l o b j e c t i v e , the descr ip t ion 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 ampl i f ied i n Chapter 2. Much of th i s material i s ava i lab le 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 into a number of subsidiary cases. The process of re -construct ion leads to a new descr ipt ion of system s t ruc ture , and a ra t iona 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 poss ib le . 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 pract ice the small d i v i s o r problem indicates 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 recent ly been developed to overcome r e s t r i c t i o n s of t h i s nature, and a two-var iable expansion procedure i s applied i n the present analys is to solve the nonlinear equations of three and four-body motion. The der iva t ion of a unique u n i f o r m l y - v a l i d so lu t ion of the s t e l l a r three-body problem i s described i n Chapter 4, and i n Chapter 5 re lated solut ions of the r e s t r i c t e d four-body problem are determined for motion i n the v i c i n i t y of and Lg. These l i t e r a l solut ions are compared with numerically-generated per iodic o r b i t s for two d i s t i n c t primary models, one corresponding to the sun-earth-moon conf igurat ion of our so lar system, the other associated with a problem of s t e l l a r motion. 9 2- 1 2. PERIPHERAL MATERIAL 2.1 Introduction ' Before we embark on a deta i led study of the three and four-body problems, some general perspective of c e l e s t i a l mechanics i s d e s i r a b l e . Six topics of p a r t i c u l a r importance are co l l ec ted here, which serve both to introduce notation and to indicate some of the basic structure of the n-body problem. A number of standard der ivat ions 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 are , i n t h i s case, indicated i n the tex t . The phi losophical basis of Newtonian dynamics i s b r i e f l y considered i n Section Z.Z, witn the in tent ion 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 introduc-t i o n to some general features of the n-body problem, and by a der iva t ion 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 in tegra l s i n ro ta t ing systems of reference. 2.2 L imita t ions and Res t r i c t ions Phenomenon: A thing that appears, or is perceived or observed; applied chiefly to a fact or occurrence, the cause of which is in question. [Oxford] Throughout the subsequent analysis 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. The ent i re representation i s , however,' mathematical. As such i t should possess complete in terna l con-s i s tency , but th i s i s no guarantee that theoret ica l resul t s w i l l agree with observable r e a l i t y . The extent of any such agreement i s determined by the mathematical descr ipt ion of physical phenomena, and a l l s i m p l i f y -ing assumptions w i l l lead to imprecision i n the process of representa-t i o n . We therefore s tar t th i s inves t iga t ion 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 analys is takes as i t s foundation the laws of Newtonian mechanics. These are [ 3 7 1 ] : Law I . Every body continues i n i t s state of r e s t , or of uniform motion i n a r i g h t l i n e , unless i t i s compelled to change that state by forces impressed upon i t . Law I I . 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 act ion there i s always opposed an equal reac t ion : o r , the mutual actions of two bodies upon each other are always equal , and directed to contrary par ts . In the discussion below, these laws w i l l be referred to as the law of i n e r t i a , motion and equi l ibr ium respec t ive ly . I t i s necessary, i f we are to avoid a lengthy divers ion into philosophy (and, to some extent, h is tory) to accept cer ta in phenomena as being commonly understood. The concepts of r e a l i t y , p o i n t , s t ra ight (or r i g h t ) l i n e , plane, space, 11 2.3 time, distance and force w i l l therefore not be questioned here. . Within a local region of the universe the laws of Newtonian mechanics generally produce theoret ica l resul ts i n excel lent agreement with observation, but we should be aware at the outset of t h e i r impl i ca t ions . 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 inconsis tent . In order to locate a s p e c i f i c event i n time and space, the observer must select a su i tab le time o r i g i n and reference coordinate system. I f the system of reference i s d i f f e r e n t from his own, however, there i s no guarantee that an event which he observes w i l l appear i d e n t i c a l in the other system Of reference. This subtle con-s iderat ion i s int imately connected with the process of transforming an event i n one system of reference into a corresponding event i n another system. Such a transformation a f fec t s not only the p o s i t i o n , but a lso the time instant 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 ther 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 of t h i s assumption i s that time af fects a l l space equa l ly , a condit ion which Newton defined as "absolute" time [ 3 7 2 ] . The properties of space are i m p l i c i t in 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 cer ta in special proper t ies ; 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 Eucl idean, for 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 in which the standards of time and distance measure-ment 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 of instantaneous t r a n s i t i o n between systems i s e n t i r e l y l eg i t imate . Two points can therefore be transformed instantaneously from one system into another without a f f e c t -ing the length which they def ine. Any r e l a t i v e motion of the two systems w i l l have no e f fec t on the ' transformation, but only because the t r a n s i t i o n i s 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 in to an equivalent sequence with respect to the other system of reference. I f 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 transform into a (d i f fe rent ) uniform v e l o c i t y i n the other system. Consequent ly , . 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 lso 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 , deviat ions from the law of i n e r t i a w i l l be observed. This immediately implies tha t , somewhere i n space, there ex 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 th i s absolute Euclidean space. Newton's law of motion introduces, i n addi t ion to time and dis tance , the concepts of force and mass. These two phenomena are 13 2.5 related by an a l ternat ive statement of the law of motion: a force F act ing on a p a r t i c l e produces an accelerat ion d i r e c t l y proportional to that force . 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 ! enon i f the law of motion i s to be v a l i d wi th in the observer's system of reference. This d e f i n i t i o n of mass can be applied to determine a further quant i ty , impulse, using the law of motion. We have, i n the d i r e c t i o n of the force F: where pndv i s the change in momentum, and the derived quanti ty 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 of 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 zero, the t o t a l change i n momentum w i l l also be zero, and therefore momentum i s conserved during the c o l l i s i o n . Now mass time and distance must be invar iant 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 are required by the law of conservation of momentum, even though i t i s derived from the law of motion. Therein l i e s i t s s ign i f i cance i n r e l a t i y i s t i c dynamics. • 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 resistance o r "mass" of the p a r t i c l e , which must be an invariant.phenom-F = r*v •Lfc and hence I l l 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 invar iant phenomenon. In general , transformations which preserve Newton's laws of i n e r t i a ! motion are of the form | * = f L J + S. t + ^ , ( 2 . 1 ) where ft i s a constant rotat ion matr ix , a i s a constant vector def in ing 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 der ivat ion of th 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 xed i n e r t i a ! reference must e x i s t . 3. Mass, time and distance are fundamental, invar iant phenomena. These abstract , but s i g n i f i c a n t , l i m i t a t i o n s inf luence the ent i re a n a l y s i s . The remaining r e s t r i c t i o n s 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 i n the study of 2-body motion. For more than two bodies, except i n cer ta in special instances , l i t t l e progress 15 2.7 appears to be possible without i t . (See also the discussion on pages 2.25 and 2.26 ). ( i i ) Only grav i ta t iona l f i e l d s r e s u l t i n g from the masses under consideration a f fec t the motion of these bodies. The e f fec ts of radia t ion pressure, t i d a l i n t e r a c t i o n , g r a v i t a -t iona l f i e l d s of d is tant masses, l i b r a t i o n a l motion and a l l other complicating features w i l l be neglected, ( i i i ) A l l the masses under consideration are r i g i d , spherical bodies whose concentric layers are homogeneous. Each may then be treated as a point mass, and Newton's law of g r a v i t a t i o n can be applied 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 separating them, the inaccuracies introduced by th i s assumption are n e g l i g i b l e . This topic i s covered i n d e t a i l by Danby [ 3 9 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 potent ia 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 res t s . E ins te in disproved Newton's basic assump-t i o n concerning the independence of time and space with the theory of special r e l a t i v i t y ; general r e l a t i v i t y further disproved the existence of absolute space, and indicated that space-time i s not general ly a Euclidean continuum. The success of Newtonian mechanics as a viable theoret ica l basis rests , - however, on the general accuracy of i t s p r e d i c t i o n 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 possible to detect 16 2.8 s i g n i f i c a n t contradic t ion between theory and observation. Neither con-s iderat ion i s l i k e l y to a f fec t the present a n a l y s i s . 2.3 N-body Dynamics The n-body (or , more accurate ly , n - p a r t i c l e ) 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 instant t h e i r pos i t ion and v e l o c i t y vectors are known, and the subsequent motion is to be determined. This problem has not been solved. Indeed, the n-body problem i s so res i s tant to analysis that a general so lut ion ex i s t s only for 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 in tegrat ion are necessary f o r a cOiiifJicoc 5 u l u c i 0 u uu t o n l y ceil i iave oeeii TOunu. C c i " i u l i i o p e o l f ' i C CuSco are , nevertheless, amenable to a n a l y s i s ; of these, the homographic solut ions considered i n Section 2.4 form an important c l a s s . The " c l a s s i c a l " integrals are fundamental to any study of these special conf igurat ions , and we therefore give a b r i e f der ivat ion of the ten known integra ls i n th i s sec t ion . Newton's law of g r a v i t a t i o n can be expressed in the fo l lowing form: There are observed phenomena between two bodies i n space which can be described by presuming that two bodies a t t r a c t 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 inverse ly 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 th i s law and consider the i n i t i a l conf igurat ion shown 17 Figure 2-1 I n i t i a l conf igurat ion f o r the n-body problem i n Figure 2-1, where we assume there ex 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 in absolute space. The motion of p^ , the i ' t h p a r t i c l e , then s a t i s f i e s the equation: 18 2.10 where r^. = r^ - r . . , i s the mass of and k denotes the Gaussian constant of g r a v i t a t i o n . An existence condit ion for solut ions of th i s equation, due to Painleve, may be found i n P o l l a r d [41. 1]; a proof appears i n Wintner [ 3 8 2 ] . Let the vectors r . , r . be given at some time instant 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. I f r ( t ) denotes the smallest of the distances r . . at time t , then there ex is ts a unique set of n vector functions r\(t) and a largest in terva l of time - t 2 < t < t-j containing the time instant t = 0 such that : ( 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 2 < t < t-j; ( i i ) r\(t) and r\(t) agree with the i n i t i a l data when t = 0. Also / i i i \ -J -f 4-l-»V% -Iri+A wi/^ 1 . 4* y 4- S" +• *?c n n + +• l-»r» i w t r > v \ » a l ^ r * - i V 4- <* m \ 4 I t y i • V l l ^ . » i t <r<~ I K U l S , ^ ^ w j • ^  . • ^  w « i < > - ' I . t v i . «^ . - t. ^ y then r ( t ) -> 0 as t t-j i f t-j i s f i n i t e and r ( t ) •+ 0 as t -> - t 2 i f t 2 i s f i n i t e . The essence of th i s theorem i s contained i n i t s f i n a l s e c t i o n , which implies that a continuous s o l u t i o n ex is t s during a time i n t e r v a l determined by the condit ion r(t) ->- 0. We s h a l l consequently r e q u i r e , for the remainder of th 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 stat ing, that a c o l l i s -ion occurs when r -* 0. The d i f f i c u l t y i s that r ( t ) = Min { r 1 2 , r ^ , • • — , ) n } may tend to zero when none of the i n d i v i d u a l distances tends to zero , 19 2.11 the ro le of l eas t distance being exchanged between them i n f i n i t e l y o f ten . To prove that the condit ion r{t) -> 0 corresponds to a c o l l i s i o n at some f i n i t e time instant t * i t i s necessary to show that the i n d i v i d u a l distances tend to l i m i t s as t -> t * , and th i s has not been proved f o r n > 3. A discussion of th i s enigma can be found i n Wintner [38 3 ] . The c l a s s i c a l integra ls can r e a d i l y be obtained from the equations of motion. Performing the sum over i we obta in : 1=1 The double sum over i and j i s ev ident ly zero, as r\ . + r.^ = 0, and therefore fV -1 \ * : F, = O . Z i (,r. I This can at once be integrated to give — ~ \ t i l-1 where a and b are constant vectors . The time i n t e r v a l - t 2 < t < t-j y def ining so lut ion existence w i l l not , i n general , be re-s tated a f t e r th i s po in t , but should be considered i m p l i c i t i n the a n a l y s i s . Now i f r defines the centre of mass of the n p a r t i c l e s : 1 I S. I 20 From Equations 2.5 and 2.6 we obtain the condit ion for conservation of l i n e a r momentum: 2.12 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 of 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 . In t h i s case a = 6 = 0, corresponding to the condit ion r ( t ) = 0, and also \ / - t , < t ^ t , . With a and b def ined, 6n -6 constants of in tegra t ion remain to be determined. We now introduce a force funct ion F by the equation n. I f f = f ( r , , r 9 , r ) , the operation - — i s defined by 3r„ ( z . i ) 5f t U f , V . where a „ , 3,,, Yi/ a r e the components of i n a Cartesian coordinate system. Equation 2.2 may now be wr i t ten i n the fo l lowing form: 21 2.13 Taking the scalar product with r\ and summing over i «. s 1 and therefore Z . cLt cC - I * "* 1 d — Equation 2.11 can be integrated, since r. . ? i = ^ 7ft ( r i * V ' t 0 g i v e the energy integral where S*, the t o t a l energy, i s an invar iant quant i ty . The energy constant S* reduces the number of undetermined in tegra l s to 6n - 7- 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 zero, since r. x ?..•= - r . . x ? i , and therefore 22 2.14 Consequently I X , where h * i s a constant vector def ining the angular momentum. The condit ion 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) . Suppose a plane IT contains the n p a r t i c l e s during the time in terva l - t 2 < t < ty The corresponding so lut ion of Equation 2.3 i s then known as planar i f the pos i t ion of IT i n the i n e r t i a l coordinate system does not depend on t , and f l a t i f IT = i r ( t ) . I f a so lut ion i s planar , then TT i s the invar iable plane, provided h * f 0; the proof appears in Wintner [38"] . The three in tegra ls associated with h complete the known in tegra l s of the n-body problem, leaving 6n-10 constants undetermined. An exhaustive treatment of t h i s ent i re subject may be found i n Wintner [385 ] . For a less deta i led a n a l y s i s , see P o l l a r d [41 2 ] and also Danby [ 3 9 2 ] . 2.4 Homographic Solutions A so lut ion of the n-body problem i s c a l l e d homographic i f the conf igurat ion formed by the n p a r t i c l e s at a given time instant t moves, with respect to the i n e r t i a l coordinate system, i n such a way that the conf igurat ion i s preserved as t v a r i e s . I f , i n i t i a l l y , t = 0, then a homographic so lut ion w i l l be of the form r - ( b ) = pCb) H C O F C O ) , L = l , 2 , r - , 23 2.15 where p(t) i s a s ca lar , fi(t) denotes a rota t ion 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. There are two l i m i t i n g cases: ( i ) i f the configurat ion d i l a t e s without rota t ion so that r L ( f c ) = ^(fc") r ( o ) ( r \l2i , the so lut ion i s c a l l e d homothetic; ( i i ) i f , conversely, the configurat ion i s . rotat ing without d i l a t i o n , then which defines a so lut ion of r e l a t i v e e q u i l i b r i u m . Three resul t s are of p a r t i c u l a r importance: ( i ) a homographic so lut ion which i s not f l a t must be homothetic; ( i i ) i f f l a t , a homographic s o l u t i o n must also be planar ; also ( i i i ) a homographic so lut ion i s a so lu t ion of r e l a t i v e equi l ibr ium only i f i t i s planar and rotates with 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 6 L Solutions of the two body problem are a l l homographic and, unless the angular momentum vector h * = 0, planar ; Lagrange's e q u i l i b r i u m solut ions of the three-body problem are both homographic and planar . The s ign i f i cance of homographic solut ions i s evident from these two cases, which together form the foundation of theoret ica l astronomy. 211 2.16 2.5 Two-body Motion. A so lut ion of the problem of two bodies appears i n most texts on c e l e s t i a l mechanics. Many resu l t s from the two-body problem w i l l , however, be needed l a t e r in the a n a l y s i s ; for the sake of completeness, therefore, the elegant and concise development given i n Danby [39 3 ] i s out l ined 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 set t ing n = 2 i n Equation 2.8: r*,F, + = o (z. is) We assume that neither mass i s i d e n t i c a l l y equal to zero. Then, from Equation 2 .2 , and using Equation 2.14: r 3 and therefore , i f r = ? 2 - r^ f = - U a H F (x. .?) where M = + m 2 . Taking the vector product of f with Equation 2.17: 25 2.17 and i n t e g r a t i n g : r x r-(%'. 18") where h i s a constant vector , 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 equivalent . Expand-ing Equation 2.18 we obta in : 1 1 ' which can be rearranged i n the form r * r-!1 +.. *s i-,*r, 1 <-s ince , from Equation 2.14, and h * i s then given by w , , ^ . The r e l a t i o n s h i p between h M From Equation 2.18, taking the sca lar 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 so lut ion i s possible because h i s normal to the invar iab le plane. Now take the vector product of h with Equation 2.17, using h x r = (r x r) x r from Equation 2.18. We obtain = - k = - w ^  H d. f r or Integrating t h i s expression t * f = - t c M f - ? ( a . n ) 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 fol lows that P . h = 0, so one constant of in tegrat ion s t i l l remains.to be found. A parametric so lut ion can, however, be found by e l iminat ing t from Equation 2.19. The scalar 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 , th i s equation can be rearranged i n the form u p ^ h 2 where -p— = e and -p— = p . ITM ITM This i s the equation of a conic with the o r i g i n at one focus; for an 2 2 e l l i p s e p = a ( l - e ) , and for a hyperbola p = a(e - 1) . The vector P points along the major axis of the o r b i t toward the pos i t ion of c losest 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 wri t ten i n the form ptt and from Equation 2.21 / ( 1 •+ ecoso) . v El iminat ing r from Equation 2.22 we obtain 28 2.20 but, although t h i s equation can be integrated, an addi t ional s u b s t i t u t i o n i s necessary to put the resu l t i n a useful form. Two cases can, however, be integrated d i r e c t l y . The c i r c u l a r o r b i t case (e = 0) presents no d i f f i c u l t y ; when e = 1 Equation 2.32 can be wr i t ten 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 in tegra t ion for the parabol ic case, corresponds to the time instant when 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 wri te F = hV •, and consequently, fo l lowing the same procedure as i n Section 2 .2 : 29 2.21 ± r . r 2- r where S i s an energy i n t e g r a l . The corresponding expression derived from Equation 2.12 i s r (2.2.?) where the two energy integra ls S and S* are re lated by n The constant S can.be evaluated from the condit ion r*. r -V At. ( 2 . 2 . S ) After subs t i tu t ing for ^ from Equation 2.22, for e l l i p t i c o r b i t s we obtain %$ « U r M 0,0 -e*) --.2 + r.. and, since r = 0 when r - a( l ± e ) , Equation 2.26 may be wr i t t en r 2. - _l_ r (a-*<0 If Equation 2.18 i s squared: 30 C = (F.F)(F. r ) - (r.F) (..*..3.o.) which can be expressed in the form k M r 1. t. . O. r — f r where r . r i s obtained from Equation 2.29, and h from Equation 2.22. Now we define the eccentric anomaly E by r = a- ( I - e cos E ) As the true anomaly v varies from 0 to 2TT, E also varies from 0 to 2u. n i f f o v o n t i a t i n n F n n a - H n n ? 3?: which can be subst i tuted into Equation 2.31; a f ter some manipulation we obtain = .1 The eccentr ic 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 . 3 3 ) 31 2.23 Integrating t h i s expression over one complete o r b i t , the period r w i l l be ITT PL The mean motion n i s defined by P and Equation 2.33 may be wr i t ten a i t = ( i - e<.c*E) crf-E This can be integrated d i r e c t l y to give C 2 . 3 O (2 .3<r) n. C t - T ) = £ - e,s^ E where T, the time instant at which r i s a minimum, i s the f i n a l constant of integrat ion for the e l l i p t i c o r b i t ; a s i m i l a r so lut ion may be found for 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 n. C23?) The preceding analysis i s based on the assumption that h f 0. I f , however, the angular momentum h i s zero, then from Equation 2.18 the motion must be one-dimensional. Although s t i l l homographic, t h i s i s the one homothetic so lu t ion of the two-body problem. An analysis of 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 regular-i z a t i o n , can be found i n Szebehely [ l 3 ] . 2.6 Non- iner t ia l Systems of Reference From t h i s point we shal l be concerned with the motion of one p a r t i c l e under the grav i ta t iona l inf luence of the remaining n-1 bodies. The equation of motion for t h i s p a r t i c l e with respect to the i n e r t i a l coordinate system may be wri t ten i n the fo l lowing dimensionless form n. ~l a .t Z Z . l r - . . | 3 J ^ where p-j = kzm.- ty3 , \ i s the distance normalization and T the time normalizat ion. We w i l l continue to use the convenient notation p^jr = r 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 considerat ion i n t h i s analys is i s subject to these condi t ions : 1. m1 > m,, > m 3 » m^ ; 2. the mass of p^ i s so small that i t has no e f fec t on motion of the primary bodies p^ , p 2 and p 3 ; We assume, i n t h i s formulat ion, 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 zero, in which case the d i v i s i o n by mi i n Equation 2.2 i s l eg i t imate . 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 pr imaries ; 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 ' f i ( 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 fourth assumption concerning motion of the centre of mass of p 2 and p-j needs some j u s t i f i -ca t ion . Tf v>* Ar>nr\-t-r\c nnc i t inn \iar- + nv r>-f n with rocnort tn r>* . r - - . • _ - r , ... • , then',- i n the notation of t h i s analys is and with respect to the i n e r t i a l system of reference; where P_ (co S 5 ) = l\lcoS S - 1 1 and cos S = F „ . r * [ 4 2 1 ] . xi J — The f i r s t term of the force funct ion corresponds to the two-body s i t u a t i o n (see Equation 2.25) ; 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 in the force funct ion i s given by + ny)x \ r* ) The worst case resul t s from m9 = m q , and with the distance normalization X* <?F 3k chosen so that ^ 3 = 1 we obtain f> £ ^ . In pract ice p 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 conf igurat ion of the three primaries i s shown i n Figure 2-2. S,H,Z form an orthogonal coordinate system rotat ing at a normalized angular v e l o c i t y Q with respect to the i n e r t i a l X , Y , Z system, with E,H i n the o r b i t a l plane of the centre of mass of p 2 and p 3 - The £,n,C coordinate system rotates at an angular v e l o c i t y u> with respect to the H,H,Z system, with £,n i n the o r b i t a l plane of p 2 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 pos i t ion vectors a , 6 and y define the same point i n the X , Y , Z , E,H,Z and £,n,C coordinate systems respec-t i v e l y . Then: where C i s the pos i t ion vector of B (the o r i g i n of the £,n,C coordinate system) i n the rotat ing E,H,Z system. S i m i l a r l y : 36 2.28 OC (w + .n.) %y + [ C ] + Z" a > <[ C']"HZ + ^L'<^-a * C ^ + * c I f r i s the pos i t ion vector of p 4 in the £ ,n ,£ system, the equation of motion for p^ may be obtained from Equation 2.38 as: r -*- 2-(l3 +st)x r + ux(unf) - f - Z ^ i - K ^ ^ r ) +• -^(xL* f) + ( i r 11- . i 3 L ** J where Equation 2.41 i s used to express r in 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 fo l lowing re la t ionsh ips : . A X cos <p — y' o A y cos y> o A z. • o o i A 2 0-43) and 37 2.29 A z ( t o s < j f c c o £ ^ + S ^ H ^ S i n . ^ ' t o S ' L ) ( - S i r t ( j £ u > S ^ + C o S ^ S i w ^ W u ) - S ^ ^ S i w V where f l = H z , n. = n £ t COS V A (2.4-4-) and i denotes the angle between the E, H and £ , n planes( Figure 2-2) The X, E and 5 axes coincide at the time o r i g i n t = 0. 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, then with respect to the E, H, Z coordinate system: Z -flay t t Sl^St^C.) t A x C = [ c . ~ JT-aC-] £ + [ -n- c] I (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, possible 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 reference; 4. 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 wr i t ten = -si c 3 _ N As a consequence of assumption 5 above, one possible so lu t ion Equation 2.49 i s 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 po in t , to choose the time normalization so that n = 1. I f we denote dimensioned quant i t ies 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 . Now i f Equation 2.37 i s used to determine the mean distance r 2 3 * : U xtr PA j . +• rrt. 3 2-3 (2.SZ) where r 2 3 * = A r 2 3 But we also have and so Equation 2.52 may be wr i t ten n. 1 3 = 1 I f , therefore , the distance normalization A.is chosen so that X = r ^ then r 2 3 = 1 and also ho 2.32 ^ + p-3 - i . a - ^O These normalized values are consistent with those i n Szebehely [ i * ] . 2.7 Integrals in 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 scalar f u n c t i o n , then G i s an in tegra l of Equation 2.55 i f i f f r ( f c . t ) l = O' ll-Sb) for a l l x , t s a t i s f y i n g Equation 2.55. The energy in tegra l of the n-body problem (Equation 2.12) i s of t h i s form, where d(f , F , t ) = j \ *:rc.F. - . . .k N. 2. fr;1 u , i>c l r * j l In a rotat ing coordinate system the equation of motion for a s ingle p a r t i c l e p^ of non-zero mass i s : F , + ^ i i x r . •+ i i x ( A x r J + j f l x F . - (_VZL' 1 = ° > |r..p 1 i r 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. I f a force funct ion U i s defined as L i I r ; . -then Equation 2.57 may be wr i t ten . >rc . and taking the scalar product of t h i s equation with r\: Now the vectors r . i n Equation 2.58 (which define the posi t ions of the remaining n-1 p a r t i c l e s ) w i l l i n general be functions of t ime, 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 for 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 = U^- = O , (.2.61 ) a t 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 xed conf igurat ion which i s unaffected by the motion of p^. This l a s t condit ion implies that the n-1 vectors r . ( j + i ) define a J homographic so lut ion of the (n-1) body problem, and from Section 2.4 t h i s so lut ion of r e l a t i v e equi l ibr ium must be planar . Subject to these r e s t r i c t i o n s , the sca lar funct ion i f ; . ^ - C " 1 p-j + r y . [ i x * (Cxi* r,.) = constant i s an integra l of Equation 2.57, although t h i s expression i s s t r i c t l y v a l i d only when m.. i s so small that condit ion 2 above i s f u l f i l l e d . Both of these requirements are s a t i s f i e d by the assumptions of the r e s t r i c t e d problem of three bodies. The corresponding i n t e g r a l , c a l l e d the Jacobi i n t e g r a l , i s 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 invar iab le x , y plane of p^ and p^, and the time normalization i s chosen'so that ft = 1 . The Jacobian in tegra l i s d i s -cussed in Szebehely [ 1 ] , but note that Szebehely uses a modified force k3 2.35 function 11"= U. + > ^ ( p.^ throughout his a n a l y s i s . The in tegra l defined i n Wintner [38 7 ] 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 in a s p e c i f i c four-body problem. The in tegra l s do not , however, s a t i s f y Equation 2.56, and the primary model for p.|, p 2 and p^ i s not a consistent so lut ion of the three-body problem. Integrals of Equation 2.57 can be determined for the four-body case only i f Equation 2.61 i s v a l i d . This condit ion i s s a t i s f i e d by Lagrange's homographic so lut ions of r e l a t i v e equi l ibr ium for the three primary masses, and corresponding surfaces of zero v e l o c i t y for the motion of a fourth p a r t i c l e of i n f i n i t e s i m a l mass have been determined by Matas. The f i r s t paper [43] presents surfaces of zero v e l o c i t y for the equi la te ra l t r i a n g l e conf igura-t i o n and cer ta in s p e c i f i c values of primary mass; a l a t e r paper [44 ] extends these resul t s to include the c o l l i n e a r conf igura t ion . Note that surfaces of zero v e l o c i t y for the r e s t r i c t e d four-body problem were f i r s t determined by MouKon [ 4 ] , although s p e c i f i c r e s u l t s were not given. 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. LIMIT PROCESS CONSIDERATIONS 3.1 Introduction For the remainder of t h i s analys is we shal 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 3 must, how-ever, be known before attempting to solve these equations. Following 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 analysis on the fo l lowing two assumptions: ( i ) p-j and p* move i n two-body orb i t s with respect to the i n e r t i a l frame of reference; ( i i ) 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 leas ingly uncomplicated, i t neglects the grav i ta t iona l i n t e r a c t i o n of p^ and p 2 and i s not a v a l i d so lu t ion of the three-body problem. 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 analys is 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 ] , [ 3 1 ] but unfortunately these precise solut ions introduce into the equations of motion a var ie ty of parameters which tend to obscure the process of analysis and made i t increas ingly d i f f i c u l t to separate one perturbation from another. 1|5 The four-body problem i s usual ly treated as a perturbed version of the r e s t r i c t e d problem involv ing p 2 , p 3 and p^ 5 so i t i s not unreason-able to expect that Lagrange p o i n t s , which are of such s ign i f i cance i n 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 two papers by Matas [ 43 , 44 ] on zero-v e l o c i t y surfaces, Shi and Eckste in 's paper [ 2 6 ] on Earth-Moon t r a j e c t o r i e s and a re lated 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 2 and p 3 . A v a l i d so lut ion of the four-body problem can only be obtained i f we f i r s t solve the three-body problem for p-j, p 2 and p 3 and then derive a corresponding so lut ion for the motion of p^. I n t u i t i v e l y i t seems that some "simplest" form of th i s four-body problem must e x i s t , probably involv ing o r b i t s of p 2 and p 3 that are approximately c i r c u l a r and where p^ moves near the Lagrange points of p 2 and p 3 - Three questions have not, however, been resolved. 1. What i s the l i m i t process ( i f any) which reduces the three-body problem for p-j, p 2 and p 3 to a two-body problem involv ing p 2 and p 3 ? 2. What l i m i t process reduces the four-body problem to a three-body problem involv ing p 2 , p 3 and p^? 3. What i s equivalent , 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 for a d e f i n i t i v e answer, but the s o l u t i o n i s indicated in 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 conf igurat ion which we s h a l l consider here i s determined by the condi t ions : ( i ) p* and p^ move in 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 i ) the three bodies p^, p^ and p^ move i n coplanar o r b i t s ; ( i i i ) 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 for p^ with 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 condit ion 2 on page 2 .24 ) . Condition ( i ) above implies that fi = C = C = 0, and we may a r b i t r a r i l y se lect <L = 0. Equation 2.42 then s i m p l i f i e s , and i n the n o n - i n e r t i a l £ , n , £ coordinate system Equation 3.1 takes the form: which can be wri t ten as: 3.4 where the time normalization has been chosen so that n = 1. The conf igurat ion 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 . In general p 2 and p^ may be expected to move wi th in bounded regions of the £ , n plane, and we fo l low Szebehely [ l 6 J i n loca t ing the la rger mass m2 so that £ 9 > 0 (Figure 3-1). 14-8 I f the mean distance between p 2 and p 3 i s chosen to be 1, then (from Equation 2.54) u 3 = 1 - i ^ . Now the centre of mass of p 2 and p 3 i s permanently located at the o r i g i n of the £ , n axes, and consequently 3.5 X " x -0 - / O m2 y 2 since — = — . From these two equations: m 3 u 3 and we may therefore wri te 1 3 « I _ L _ - 1 3 k9 The equations of motion for p 2 now take the form 3.6 S ' Z t + 5 l r , z-3 = nzc tosC\-.o.)t + |x, - g") C3 ' 2 - ) 1 + H + 1 ( 3 - I 3 ) where, f o r convenience, the subscript on £ 2 a n a " n 2 n a s ' 3 e e n o m i t t e d . These two equations are s u f f i c i e n t to determine the motion of both p 2 and p^» since £ 2 a n d ^3 c a n be found from Equations 3.5 and 3.6 once £ 2 and n 2 are known. We can, moreover, r e a d i l y demonstrate that as the two quant i t ies and 'Ml tend to zero , Equations 3.12 and 3.13 reduce to the equations of two-body motion i n a coordinate system rota t ing with constant angular v e l o c i t y . In the two-body case involv ing p 2 and p 3 , the motion of p 2 i n the ro ta t ing £ ,n ,£ coordinate system i s determined by = o , (1.1*") 5P 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 of ' re ference . This equation may be wr i t ten 3.7 jx 4 z 5 ^ - Ix + ( / i x - * t j « O (I. ' O where i t i s assumed n = I . Equations 3.10 and 3.II are s t i l l v a l i d i n the present case, and so the motion of p 2 i s defined by the equations i + s I — - 1 1 + t I - 1 J- 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 invest igate the l i m i t process fi C -»- 0 and I U 3 a s t e r i s k , then I f dimensioned quant i t ies are denoted by an 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 condit ion 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: JTV- k V (3- zi ) where i s the normalized distance between p^ and the hypothetical p a r t i c l e p* . 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 condit ion , C r , - C ) = ( w i 3 ) C , so that (3 . ZZ> From Equations 3.21 and 3.22 we can now wri te r C 3 . 2 3 ) where the quanti ty k 1 " ^ 1 / 3 i s completely determined by the conf igurat ion of p 2 and p 3 . I f Equation 3.23 tends to zero with T and X f i x e d , then e i ther m-j 0 or r^ <=°. The f i r s t condit ion i s inconsis tent with the 5 2 + 2 requirement that m^  > , and therefore ft C -> 0 must be equivalent to 3.9 the process r-j -> °°. Now and r,. r, - X t-, . r f c + r^.. F t , so that r»\, 1 + • L 0 — ( 3 . 2 4 ) which i s a form s i m i l a r to Equation 3.23. The same argument can conse-quently be applied when Equation 3.24 tends to zero , provided the term -1 i s well-behaved as r-j -»• °°. This condi t ion i s s a t i s f i e d when r 2 i s bounded and r 2 < r i » which ">s c e r t a i n l y true in the present s i t u a t i o n . Question 1 on page 3.2 can now be answered as f o l l o w s : the three-body problem f o r p-j, p 2 and p 3 , defined by Equations 3.12 and 3.13, can be reduced to a two-body problem involv ing p 2 and p 3 by a l i m i t process of the form e -*• 0, where z -»• 0 as r^ -»• » . We s h a l l , i n f a c t , take ^Condition T on page 2.24 states that m-j > m 2 > m 3 » m^. 53 3.10 e = / 1/ \ 1 , but t h i s choice w i l l be discussed l a t e r i n Section 3.4. 3.3 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 f o r the four-body case and Equations 3.3 and 3.4 f o r three-body motion, i t might be ant ic ipated that the same l i m i t process can be appl ied to both sets of equations. To demonstrate the t ruth of t h i s conjecture, consider the motion of p^ in the ro ta t ing £ , n , t, coordinate system under the g r a v i t a t i o n a l influence of p 2 and Py By d i r e c t analogy with Equations 3.14, 3.15 and 3.16 we may w r i t e : i r * * r 3 A- o where £> n define the motion of p^. Equations 2.47 and 2.48 reduce to Equations 3.25 and 3.26 when the terms and 5k tend to zero. This condit ion i s i d e n t i c a l with that derived in 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 in the four-body case. This reduction of the three and four-body problems can be interpreted i n the fo l lowing way: as p-j becomes more d i s t a n t , i t can be assumed with increasing accuracy that ( i ) the motion of p 2 and p^ i s determined by two-body dynamics, and ( i i ) the motion of p 2 , p 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 i s f i n i t e , since terms involv ing m^  (through the quant i t ies 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 i s 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 der iva t ion provides some j u s t i f i c a t i o n for neglecting the g r a v i t a t i o n a l e f f e c t of remotely d is tant masses (see r e s t r i c t i o n ( i i ) on page 2 .7 ) . Suppose, for example, that m 2 and m^ correspond to the earth and moon mass 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. In t h i s case m 2 + * m 2 , so that Equation 2.52 may be wr i t ten and, i f n = r ^ = 1, we obtain k 7 " ^ =£: i rvt.. «-3 55 3.12 From Equations 2.43 and 2.44 the pos i t ion 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- XL ) t I f these values are substi tuted into Equations 3.12 and 3.13 a cance l la t ion occurs (Appendix 1, Equations A1.4 and A 1 . 5 ) , and the leading term involv ing m^  takes the form k ^ r * , 1 - . . . . A f t e r s ubs t i tu t ing for k1" t1/ 3 we may write m l 17 For the case out l ined above (very approximately) —- = 0.4 x 10 and 11+ 2 r 1 = 8 x 10 , so that the c o e f f i c i e n t of the largest d i s turb ing term i s 0.8 x 1 0 " 1 9 . +From A l l e n [45 ] : 20 Sun's distance from ga lac t i c centre = 10 kpc = 3.0857 x 10 m. Galac t ic mass = 1.4 x 10 1 1 m Q m o 6 where ni denotes the so lar mass. We have — * 0.3 x 10 o m 2 m T 17 (from page 4.37), and therefore — - 0.4 x 10 . The distance normaliz-2 ation A = 0.3844 x 109m (page 4 . 3 7 ) , „so that r 1 - 8 x 1 0 1 1 . 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. The d i f f i c u l t y i s associated with the terms involv ing cos(l-£2)t and s i n ( l - f t ) t in the equations of motion (2.47, 2.48, 3.12 and 3.13), but i t can be resolved quite e legantly by choosing a new time var iable t 4 = (\-Sh) t . (3.2?) From page 2.18 ^ , so that i f we set (3 -zs ) Equation 3.27 may be wr i t ten t + = [ I - e 3 ( n - j O * - ] t . . (3.21) We sha l l immediately drop the notation t + , and from t h i s point i n the analysis t , the independent v a r i a b l e , i s re lated to dimensioned time t * by f « 0 - . a ) f c * / = C l - J ^ ' t O t * , = ( a * - J C l * ) t . (3 .3a) ' X / "17 With respect to the new independent v a r i a b l e , Equations 3.12 and 3.13 .take the form =. k ^ t ^ m , , and from Equation 3.21 S1L -\ 3 r ,» 57 3.14 I - 1 ( 3 . 3 1 ) =• c o t + ^Xti-S) r • . . ^ J _ - 1 (5.32.) where we have used the condit ion SL^C = u, / t from Equation 3.23, and £ , n define the motion of p 2 . S i m i l a r l y , for the four-body problem, Equations 2.47 and 2.48 become 58 3.15 where, i n t h i s case, £ , n define the motion of p^ (the three and four-body cases w i l l be treated separately, so there i s no r i s k of confusion) . 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 (*{/ % cance l , so that both 2k 1 expressions involve sequences i n e , 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, in cer ta in cases, can be solved exac t ly . These special cases w i l l now be invest igated 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 - Now although the two-body problem can be solved (Section 2 .5 , Equations 2.32 and 2.36) the so lu t ion 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 so lut ion 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 zero , and Equations 3.17 and 3.18 become 59 3.16 ( 3 . ) 1 ( 3 3 6 ) where the expression J _ - 1 has been developed using Equation 3.9. The p a r t i c l e p 2 must therefore be located somewhere on the c i r c l e S % ^ 1 = ( 1 - ^ J 1 OS?) For convenience we may take n = 0, so that and, from Equations 3.5 and 3.6: (3.3<0 I f the posi t ions of p 2 and p 3 are defined by Equations 3.38 and 3.39, then homographic equi l ibr ium solut ions 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). Set t ing the d e r i v a t i v e terms equal to zero i n Equations 3.25 and 3.26: 60 3.17 = o ",1. '<f3 i = O where 5, n denote the pos i t ion of p 4 and, to fo l low Szebehely's notation [ l 7 ] , we have substi tuted y 3 = y , y 2 = 1 - y . One s o l u t i o n of Equation 3.41 i s n = 0, in which case r - <>-s.) : Equation 3.40 then s i m p l i f i e s to ± ft = o ( 3 . 4 2 ) which defines the three c o l l i n e a r Lagrange points L-j , L 2 and L 3 [ l 8 ] . The remaining so lut ion of Equation 3.41 i s 1 - ti-h> = D (1.4-3) which, when 0 < y < 1, reduces Equation 3.40 to the form ( 3 . 4 - 0 1 ' •V 61 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 , 3.18 4 3 ' = 1 C 3.4-5) defines the two equi la tera l t r i a n g l e points L^ and L^ (Figure 3-2) . An analys is of the two and three-body equations of motion (Equations 3.17, 3.18, 3.25 and 3.26) i n the v i c i n i t y of these exact solut ions i s now f e a s i b l e , but two addi t iona l small parameters, 6 and a , must f i r s t be introduced. The impl icat ions of t h i s process are considered i n the fo l lowing sec t ion . 6 2 3.19 3.6 Orbi ta l Perturbations 3 . 6 . V Perturbed 2-body motion The major obstacle preventing an e x p l i c i t so lut ion of the equations determining two and three-body motion i s the presence of non-l i n e a r terms involv ing _ ! Close to an exact s o l u t i o n , however, these terms can be expanded as functions of a small o r b i t a l parameter. For the two-body motion of p 2 t h i s exact so lut ion i s given by Equation 3.38 and, i f we take S x ( t ) = I - p . , . + l e f t ) ; iJ-t} - S f C b ) ( 3 . 4 6 ) a binomial expansion of the term _J leads to a v a r i a t i o n a l i r x 3 l 3 form of Equations 3.17 and 3.18: [ e -2i - 3 e l + L u\-fS> J - o ( 3 . 4 * ^ S [ ? + 2 e ] - 5 3ef - o ( 3 . 4 8 ) 6 3 3.20 + where 6 « 1. Now suppose we consider the addi t ional 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 fo l lowing modif icat ion 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 perturbat ion. The equations of motion are , 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 ixed they reduce to Equations 3.47 and 3.48. When 6 ->- 0 with e f ixed we obtain 4-To see t h i s , note that from Equation 3.9: and therefore: 0 - f O * *" _ S Now take _ L _ Used-pi) + V [ e ^ f x ] " ? , 0 - H O 1 <• ... J and expand (1 + r ) " 3 / z . The expressions appearing in Equations 3.47 and 3.48 fo l low 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 9 and when the only disturbance from r e l a t i v e equi l ibr ium i s caused by p-|. Consequently they define the "simplest" v a l i d three-body model for p-j, p 2 and p^, and the so lut ion of Equations 3.51 and 3.52 forms the basis of the "s implest" four-body problem. 3.6.2 Perturbed 3-body motion The e x p l i c i t solut ions for three-body motion are defined by Equations 3.42 and 3.45. I f the terms i and _j 3 * 3 ' are expanded about a p a r t i c u l a r Lagrange point nL» then with we obtain the fo l lowing 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 IV) : 65 3.22 C r [ - %y + 0, +/ j +• OC s r " 0 >= O ( 3 . 5 - 4 ) where a « 1 and the c o e f f i c i e n t s v-j, and depend on and n ^ . At L . , for example, o{ = - 3 / . , ^ = 3£? ( 1 - 2 . ^ ) and ° 3 = ~ • 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 deta i led analysis appears i n Szebehely [ i 2 ] . A more complicated s i t u a t i o n resu l t s i f p 2 and p^ move i n e l l i p t i c o r b i t s about t h e i r centre of mass. For small e c c e n t r i c i t y the motion of p ? can be described by Equation 3.46, so that _ i small parameter 6. For the expansion of these nonlinear expressions we require var iables , of the form S^/t) =• $^ + -5"e^ t:>+ t r s i ( b ) 'f Y /tr) = Y u + 3T^t> * .try f t } , where the v a r i a t i o n a l terms e(t) and f ( t ) correspond to perturbations a r i s i n g from the e l l i p t i c motion of p 2 and p^ (note that e(t ) and f ( t ) define variables d i f f e r e n t from those i n Equations 3.49 and 3 . 5 0 ) . T t This dupl i ca t ion i s unfortunate, but these functions take no fur ther part in the analysis 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 tuat ions i f the t r i v i a l case e -> 0, 6 . - 0 is excluded. 1. When 6 -> 0 with a f ixed the equations of motion reduce to Equations 3.54 and 3.55 for 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) - £ u +• SeCt) ; ^ = 7_ + S^Ct^ , O* 7) and consequently i t i s not possible to speci fy 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. A l s o , as 6 + 0, the functions 6e('t) and 6 f ( t ) must tend to zero uniformly i n t (see Section 3.7.1 for the d e f i n i t i o n of l i i l i i.Ci in i t,j'} • i i i i S C O n u i C l O i i c A c l U u c S i c C t i - l u i " t c i i n S ' f i ' C i i i t i i c so lut ion and i s s u f f i c i e n t to determine e(t) a n d ' f ( t ) uniquely. A proof of t h i s statement w i l l not be g iven , 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 equi l ibr ium solut ions 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 de-scribe 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- analys is 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 into consideration the e f fec t of py, the l i n e a r functions _J and _ ! _ w i l l contain time-non-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 perturbation caused by p^ (note again that u(t) and v( t ) define var iables d i f f e r e n t from those i n Equations 3.49 and 3.50) + then, with eight l imi t -process combinations are poss ib le . Four of these involve the process e •> 0, and consequently lead to s i tua t ions described previous ly . The remaining combinations are l i s t e d below i n increasing order of complexity. ( i ) 6 and a ->• 0 with e f i x e d . In t h i s case the motion of p-j, p 2 and p 3 i s defined by Equations 3.51 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 four-body equivalent of Lagrange equi l ibr ium solut ions of the r e s t r i c t e d problem, may be determined from the condi t ion that eu(t) and ev(t) i n Equations 3.58 and 3.59 tend to zero uniformly i n t as e ->- 0. ( i i ) a 0 with <5 and e f i x e d . In contrast to the preceding case, where p 2 and p^ move i n approximately c i r c u l a r o r b i t s ( 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 ) , (3.5-1?) C3.S"0 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 apply, and corresponding o r b i t s f o r p^ could be determined, ( i i i ) 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 define motion around the o r b i t s of case ( i ) above, ( i v ) a , 6 and e are a l l non-zero. 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 lucidate the process of evolut ion described i n t h i s s e c t i o n , XI-, • -C _ ^ T ^  4-. — I J . . „ . ~ U 4 i „ 4-„ kn>l.. .v.4--J ,-n ~ A r-U l l l ^ j ~ M l / ^ I C ; . j . J I \ > l l I I i l l V, I I I U I I , » « U "~ v> t - i j r \s t U I \**J . • n . ^ ^ . v / i . i ~i summarized in Table 3-1. An as ter i sk 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 pos i t ion and v e l o c i t y of p^. The morphosis of th i s 4-body problem i s i l l u s t r a t e d i n Figure 3-3. Interact ions between the various p a r t i c l e s are represented by f igures 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 for 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 e 6 0 Motion of p 2 and Motion of p^ 1 0 0 C i r c u l a r 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 C i r c u l a r Lagrange point 6 - 0 0 * C i r c u l a r V a r i a t i o n a l o r b i t 7 0 * 0 E l l i p t i c 3-body unique o r b i t 8 0 * * E l l i p t i c V a r i a t i o n a l o r b i t 9 * 0 0 Perturbed c i r c u l a r 4-body unique o r b i t 10 * 0 * Perturbed c i r c u l a r Var ia t iona l o r b i t 11 * * 0 Perturbed e l l i p t i c 4-body unique o r b i t 12 * * * Perturbed el 1 i p t i c V a r i a t i o n a l o r b i t Table 3-1 Orbi ta l motion in the n-bodv Droblem for 2 * n ^ 4 . equi l ibr ium solut ions of the r e s t r i c t e d problem (node 5) and with the simplest three-body model for p.|, p 2 and p^ (node 3) . The "s implest" form of the four-body problem, represented by node 9, i s the l o g i c a l consequence of these three fundamental cases. A l l that we now require i s a method by which the nonl inear , non-homogeneous equations of case 3 and case 9 can be solved. 3.7 The Two-variable Expansion Procedure Several perturbation methods have recent ly been developed to determine approximate solut ions of equations which, for a var ie ty 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 re la te to approximation methods i n general w i l l repeatedly be applied in the present a n a l y s i s , arid for the sake of completeness these are defined here. Further d e t a i l s can be found i n Nayfeh [46 ] . 3.7.1 D e f i n i t i o n s  Gauge funct ions : I f f (e) denotes a function which depends on the small parameter e, then, provided the l i m i t as e ->• 0 of f (e ) e x i s t s , when e •*• 0 we could have ( i ) f (e) + 0 ; ( i i ) f (e) ->- A where A i s f i n i t e ; ( i i i ) f (e) -> °° (or f(e) -»-•-«>) . The rate at which f(e) -*• 0 or f (e) -> °° can be expressed by comparing f(e) with a known gauge function g(e) , and of these the simplest and most useful are e e , I e , , e j • •• The comparison of f(e) with g(e) employs the Landau symbols, 0 and 0 . Large 0 I f there ex 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 wri te He) = 0 [ gCe)] as e -> O The condit ion f o r Equation 3.60 to be v a l i d can also be stated i n the form e -5>e> 5U) I f f i s a funct ion of another var iable x i n addi t ion to e, and g(x , e) i s a corresponding gauge f u n c t i o n , then as € —> O (3 .62.) i f there ex is t s a p o s i t i v e number A independent of e and an eQ > 0 such that When A and eQ are independent of x , then the condit ion defined by Equation 3.62 i s said to hold uniformly. 73 3.30 Small o: I f , f o r every pos i t ive number 6 independent of e, there e x i s t s an eQ such that I He) I £ l 3 ( e ) | for U l ± 6 then we wri te H e ) = o [ g ( e ) J as £ —> O ( 3 . 6 3 ) The def ining condit ion can be replaced by £ 0 ) = o (3. 64) which i s a form corresponding to Equation 3.61 Asymptotic Expansions: A sequence of functions <5n(e) i s c a l l e d an asymptotic sequence i f n L r»--i J as e ->• 0 (3.65) I f 5m(e) i s an asymptotic sequence and a m i s independent of e, the i n f i n i t e sum \ a S ( e ) i s the asymptotic expansion of a function HI - O f i f and. only i f 71+ 3.31 & = 2^ ^ S ^ C f e } +" ° L" as e + 0 . ( 3 . 6 6 ) We then wri te Uniform v a l i d i t y : I f x i s a scalar or vector var iable 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 am(x) are independent of e. This expansion i s uniformly v a l i d i f / . — i - o where uniformly for a l l x of i n t e r e s t . For these uniformity condit ions to be s a t i s f i e d , a m ( x ) 5 m ( E ) m u s t b e small i n comparison to the preceding term 75 3. a m _ 1 (x)6 m _ 1 (e) for each m. Now from Equation 3.65 6 (e) = o[6 ^e ' ) ] so tha t , for the expansion to be uniform, am(x) must be no more s ingular than am_-|(>0- Each term i n the expansion must consequently be a small correct ion to i t s predecessor, independent of the value of x. This concept forms the foundation of the method of mul t ip le scales [ 4 6 2 ] , of 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) and as e + 0 , C3.72) where <f>m(e) i s an asymptotic sequence. The operation of l i n e a r combin-at ion i s j u s t i f i e d i n general , and we have 00 • « K * j O + C x ; 0 ^ 1 - ^ * P" ^tS^ as e 0. ^ s e » (3.73 ) I f , however, the two expansions are formal ly 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 of 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 condit ion i s s a t i s f i e d f o r the important case <j>m(e) = e m , and we obtain 76 3. 6 (x^a) 3 C;x ; e ) A / \ ( O e " a s e -> 0 ( 3 . 7 4 ) where 2 _ K _ ^ > ( 3 . 7 S ) I f f ( x ; e) and am(x) i n Equation 3.71 are integrable functions of x , then »c as e ->• 0 so that term by term integrat ion of an asymptotic expansion i s j u s t i f i e d . When f ( x ; e) and <i>m(£). are integrable functions of e a corresponding r e s u l t holds for integrat ion 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 or e cannot, however, be j u s t i f i e d i n the general case. We cannot, consequently, assume without reservat ion 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 general , 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 into the r e s u l t i n g expansions. The basic reference for a l l the resu l t s out l ined above i s Van der Corput's 1956 paper [ 4 8 ] . Addit ional discussions appear i n Nayfeh [46 3 ] and Erdely 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 lass of methods involv ing transformations of both dependent and independent v a r i a b l e s . A deta i led descr ipt ion of these mul t ip le scale asymptotic methods can be found i n Nayfeh [46 ] , and only those aspects which apply to the present analysis are discussed here. We are interested i n der iv ing resul t s which are v a l i d over long periods of t ime, and th i s condi t ion w i l l be s a t i s f i e d i f the solut ions are uniformly v a l i d i n t over some " large" time i n t e r v a l . Consider, for example, Equations 3.51 and 3.52 which describe the three-body motion of p 2 and p^. The var iables 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 var iables x and r , so that oo u. (t^) ro \ um( t v) em as e 0 (3 .?s) L—j or L—i 78 3.35 where R n (x , T; e) = 0(e n ) uniformly i n T and r as c + 0. 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 / V m - o m 6. as e 0 . C3.8o) Corresponding expansions apply for v( t ) and ^ when v m ( x , r ) i s subst i tuted for u ( T , r ) - i n Equations 3.78 to 3.80. The t i m e - l i k e variables T and r are selected so that one, r f o r instance, i s a near - ident i ty transformation of t and the other , x , s a t i s -f i e s the condit ion x = 0(et) as z 0. For the three and four-body problems considered i n the present analysis the equations of motion contain t e x p l i c i t l y , and i t i s therefore both convenient and expedient to set the f a s t variable r equal to t . The slow var iab le i s chosen so that x = e t , which i s the simplest case s a t i s f y i n g x = 0 (e t ) . Uniformity i n x then requires that R n ( ^ , T; z) = 0(e n ) for x f i x e d as z -> 0, so that t = 0(e~^). This condit ion consequently determines, for small values of e, the required large time in terva l during which the so lu t ion i s uniformly v a l i d . The general izat ion of t h i s procedure to the many var iable case and a time in terva l 0(e~ n) (where n" = 2 , 3 , ••••) i s described i n Nayfeh [46 t f ] . When the asymptotic expansions for u(t) and v ( t ) are subst i tuted in 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 iminat ion 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 analys is of three and four-body 79 3. motion. A uniformly v a l i d so lut ion 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 so lut ion 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 so lut ion 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 for motion near the equi la te ra l t r i a n g l e points of m9 and m-. 80 4.1 4. THE THREE-BODY PROBLEM 4.1 Introduction Equations 3.51 and 3.52, together with Equations 3.5 and 3.6, describe the least-complicated form of three-body motion for p-|, p 2 and Pg, and the so lut ion of these equations w i l l be out l ined i n th i s chapter. The i n i t i a l analysis presented here i s s u f f i c i e n t to reveal the basic so lut ion s t ruc ture , 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 deta i led development i s relegated to Appendix I I I . I t w i l l , perhaps, be helpful at th i s point to summarize .the sequence .of r e s t r i c -t ions 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 2 and p 3 i s coplanar. F i n a l l y , from Section 3 .6 .1 : 81 4.2 7. As the small parameter e tends to zero , the loca t ion of P 2 i s defined by the so lut ion of r e l a t i v e equi l ibr ium C 2 ( t ) = 1 - y 2 ; n 2 ( t ) = 0 . Note that conditions 5, 6 and 7 combine to produce the simplest form of three-body motion. The only approximation i s associated wi th the second assumption. This source of inconsistency was discussed i n Section 2.6 (Equation 2.39), but as e -> 0 the r e s u l t i n g error reduces to zero. The analysis r e s t s , therefore , on a comparatively f i rm foundation. 4.2 Pre!iminary Analysis From Equations 3.49 and 3.50 the pos i t ion 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 variables x and t , with x = et , so that $JtO ^ •«-/*,.•+ -T1 u ^ r , t ) e ~ 4- OC<L^) C4 3) L — i 82 where these expressions are to be uniformly v a l i d in T and t . I f the formal operation of d i f f e r e n t i a t i o n i s assumed to be v a l i d (see Section 3 .7 .1 ) , then for the der ivat ives of eu(t) and ev(t) we obtain: 4.3 N C4.?> 6 /v/ Jib 4- & 3y„, 5" Pb^Tr Pb"1" ^ b ^ r Equations 3.51 and 3.52 can now be expressed as a sequence of non-homogeneous p a r t i a l d i f f e r e n t i a l equations i n ascending powers of e, For the i ' t h power of e we have <)t v - 2- h» 3 ^ - ^ c k £*C u,v, b, v ) ( 4 . 1 ) In the absence of any information about i t s v a l i d i t y , we have no a l t e r n a t i v e but to proceed i n the hope that t h i s operation i s j u s t i f i e d . ^t 1- at 83 4.4 .K ¥ ( u,v; b , v ) , 0*--'°) where the functions E* and F* involve ( in general) the terms u^ to u^._^ and v-| to v^_^ but not u^ and v^. Now the ordinary d i f f e r e n t i a l equations £ - ? . < / - 3u_ = o (>•") have the fo l lowing s o l u t i o n : where A, B,C and D are a r b i t r a r y constants. The homogeneous so lut ions f o r u.j ( t , x) and v^ ( t , x) i n Equations 4.9 and 4.10 therefore assume an i d e n t i c a l form except that , i n t h i s case, A , B, C and D are functions of x. Equations 4.13 and 4.14 can be rearranged so that i ^ l t / e ) = ZolL-v) - izCv)c°$[t +• 00c)] (4 .1?) which, i n p r a c t i c e , i s a more convenient form. The functions a ( x ) , 3 ( T ) , Y ( t ) and 9(x) are to be determined from the condit ions f o r uniform 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 forc ing terms involv ing Uy, Equations 3.51 and 3.52 possess the exact so lu t ion ~ ^ ~ ^ 2 ' n 2 ^ = ^ ' a s discussed i n Section 3.5. I t i s therefore reasonable to assume that any disturbance from r e l a t i v e equi l ibr ium caused by the forc ing terms w i l l be of a comparable order in e, so we can take £ <^ f c ) = J - " ^ ( t , T ) r + OUn) . (4.1?) Z > r>.-l I , . (4. 18) »~ =. k where k > 1. To determine the value of k we must now examine the non-hnmnopnpnu?; funrt.innf; and | r „ ! 3 i n Equations 3.51 and 3.52, which can be expanded using Equations A1.6 and A1.7 i n Appendix I . As the non-homogeneous functions are O'(e^), i t i s not u n r e a l i s t i c to assume that k ^ 4. From Equations A1.6 and A l . 7 we then obtain 85 4.6 + 6 3. cob JS_ 3b S (4 - . 2 ° ) since £ 2 = 1 - y 2 + ° ( e 4 ) a n d ^2 = 0 ( e 4 ) when k = 4. From Equations 4.15 and 4.16 u^( t , x) and ( t , x) are per iodic i n 8 8 t = 2TT, SO 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 so lut ion for Ug(t, x) and Vg(t , x ) . I f the complete so lut ion i s to be uniformly v a l i d in t these secular terms must be e l iminated , which can only be.accomplished i f the quant i t i es and 8 i n Equations 3.51 and 3.52 are 0 ( e ) . This condit ion i s s a t i s f i e d when k = 5. The same r e s u l t can be obtained, i n a less 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) for 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 therefore , from Equations 4.15 and 4.16: ^ ( t , f ) = 2yScO sc*.[ t + 9 C T ) ] + X C V ) - 3-bocCv) .. (*-*0 Higher order solut ions w i l l , i n general , consist of both homogeneous and non-homogeneous terms, so that we can wri te Z . — x m a t 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 . I t i s convenient to accumulate a l l the homogeneous solut ions into u 5 ( t , T ) h and v 5 ( t , T ) h , but i n t h i s case we must express the functions a(-r), B ( T ) , y("0 and 6(x) as asymptotic expansions of the form I. i = 2 ^ ^ ^ ^ ^ C + -3 I ) Equations 4.27 and 4.28 can then be wr i t ten 88 4.9 feu. Z . — i yt-l Z — - i m - 6 Now i t i s always poss ib le , by r e - l a b e l l i n g x as e t , to express the non-homogeneous solutions u . and (where i > 6) as functions of t , and therefore no der ivat ives 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 it* -a dO -5 it + E - 5. J O i l ) (4- .»0 -3u.f = -Z — -o( 0 - 2 — J Of.) 89 4.10 >Jt< it - -^ it his — r -oCfe) + E, J O ( 0 ( 4 . 3 1 ) +- 2 3 u g i t 2 0 + - p,) 1 ". [5 where the subscripts 0(1), 0(e) and 0(e ) denote terms of corresponding order in e and the functions and F. are defined i n Appendix I (Equations A1.8 and A1.9) . The extension of t h i s sequence to higher orders of e i s given i n Appendix I I I , Equations A3.20 to A3.29. 4.3 The Uniformly V a l i d Solut ion 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 condit ion that no secular or unbounded terms remain i n the s o l u t i o n . A development as far as so lut ions for Ug and Vg i s s u f f i c i e n t to reveal the essent ial features of the a n a l y s i s , and d e t a i l s of the further development can be found in 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 wri t ten i n the form &*Ctp) ~ a „ 4- a.xb 4- a ^ j U v b 4 - A ^ C s o t 4- c D « i ^ w f c + c . t o o k ( 4 . 4 l ) P * ( f c , V ) = yo + + b^<-b by l o t •*• < ^ o A V < 0 b -4- cL{ Co« o b ( 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 III (Equations A3.10 to A3.13) , and the conditions under which secular terms are el iminated 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 substi tuted in Equations 4.25 and 4.26 we obtain "" 6./», + fcl/3*.*--- J COS £ b + 0o f £0, + e?9^ir -• ] The functions cos ( t+0) and s i n ( t + 9 ) are expanded as Cn C t + P) - <^ b coP 0 - <- b «u. ©„ - «E> I s—t c<o ( 4 . 4 5 ) ( t + 6) * sCtb c«o9p 4- cr> b sJ-v. P„ + £ P, I dr»t CoPo — 1^ b P e ] + _ o f f r O so that , for the solut ion of Equations 4.23 and 4.24, we may wr i te 91 4.12 (a. 4 y ) The secular term in Equation 4.48 could be el iminated by se t t ing CXQ(X)=0, but t h i s assumes, without any addi t ional proof, that «Q(T) = constant. I t i s not , i n general , necessary to make any assumptions of t h i s nature, so we w i l l proceed as i f CCQ(X) ^ 0. Af ter 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 subs t i tu t in g for Eg and Fg from Appendix I (Equations A1.8 and A1 .9 ) , we obtain for Equations 4.35 and 4.36: 2 <-<-< Oil) - 6b Z«0 2v + 2s.«-b + 2c<rz> b ^ - x *• % SC«-b 4- <ht 5 — 2. dot ^/3„ C«Pp - p>0^^0 p_Pe I f terms in these equations are i d e n t i f i e d with quant i t ies i n Equations 4.41 and 4.42, a corresponding non-homogeneous so lut ion can then be derived from Equations A3.10 to A3.13 i n Appendix I I I . We have, for example, 92 4.13 2>V « i . [ Wo*****] ; ' 2 To el iminate terms i n v o l v i n g t from the non-homogeneous so lut ion }x0 = o (A- * > so we are now j u s t i f i e d i n taking * 0 - ° C^ .rz) to remove the secular quantity i n Equation 4.48. I f terms containing t s i n t and tcost are to be e l iminated: I P f f I =. © , (A - .SO ^ 1 so that , i f 3 Q ^ 0 , both BQ and 6Q must be constant. When Equations 4.51, 4.53 and 4.54 are s a t i s f i e d we have 93 4.14 u6 = pit1'?*-} [ X c e r > i : "<«,zt ^6 - £ * 2-t - 4jiK . t + £ t 0 Ur 0t? [ * u . t - b ] I f u c i s to be free of secular terms, then <LXo must be constant. Now 6 its contains a term -e[3ta^] which, i f a-j i s constant, could be used to el iminate the secular terms i n Equation 4.56. Suppose we ant i c ipa te the r e s u l t that = o (Equation 4 .64) . Then, i f the larges t secular terms i n the complete so lut ion are 0(e ) when t i s f i xed 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 O We wil1 then have O I 9k 4.15 .where TQ and are constant, and Equation 4.57 may be wr i t ten as 17-Exactly the same condit ion i s obtained as i n Equation 4.58, but for + r.j instead of a-p F-| is therefore redundant, and we are l e f t with YQ(T) "= r Q , which i s equivalent to Equation 4.59. The non-homogeneous solut ions Ug(t) and Vg(t) now reduce to ii $ul fc. - + £ b ? 4-so lut ion by the correct choice of For Equations 4.37 and 4.38 we have, from Equations A1.8 and A l.9 i n Appendix I : o With the s i m p l i f i c a t i o n s resu l t ing from Equations 4.51, 4.53, 4.54 and 4.59 we obtain for the d e r i v a t i v e terms: 3 u. s x V , Otl ) i v 4- sc~b 95 4.16 I at- \ i v J \ L air ar ' 5i av L a*r a_Pc _ «*p„ ) z^s, aj>0 4-I Ptr av ' ar' ^ v5 -t- - a \ s Z f^, . aT^tr a r a r P/S Zc« f f , J / J , £P, +- /5,a_P0 ? *• a v i v ) I I W J L air a r a t " A ^  - co j k a"v 1 v <aW J (. a r a-r ' a t \ ) _ Secular terms involv ing t , t s i n t and tcost w i l l be e l iminated from the solut ions for Uy and Vy i f : 1 « , = o 96 4.17 at? T v (4.6$) j-c* V a w ' a r a-r s o and these conditions are a l l s a t i s f i e d when , and 6^ are constant. The remaining secular term i n Vg i s removed i f a V atr a v at: 1 + 3^ 9 n a i r a-t"1- I a i r y where i s the only term which i s undetermined. From the same argument as before for YQ i t fol lows that A l l the quant i t ies in Equations 4.62 and 4.63 are consequently e l iminated , so that we obtain H?(fc) = V ?(t) = O . (4.61) The condit ions for 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 for ctg and , and to determine BQ and 6Q we must consider the solut ions 97 4.18 for Ug and v g . From Appendix I , Equations A1.8 and A1.9: 1 tosl + ]S COS 3fc « 8 3_s^b + If 5 ^ 3b 8 8 and the non-homogeneous terms i n Equations 4.39 and 4.40 can be wri t ten i n the form hx ix - ]£. j * , ( l - ^ . J U ' ) c o ^ S t */ 60s t i x iX e + jc*.b 2 C 0 O ^ 5 t - Zs^B0a0^K • + 2 - 0 + - / O X A > ^ " ^ o (4.?3) 98 4.19 2 To remove terms involv ing t from the non-homogeneous so lut ion ^ 2 . - 0 C 4 . ? 0 o and for the e l iminat ion of the quanti ty -e [3ta,J from Equation 4.44 we must take = 0 . (4-.7O Secular terms containing t s i n t and tcost are el iminated i f I 2V J lb and bx . 16 Now, unless s i n 6Q = 0, Equation 4.77 v io la tes 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 N a r e constant, i s also constant. I f , however, u u it 0 2 i s also constant, then i t fol lows that fi0 = is cos 9V C*-*0 Although th i s der ivat ion for gg and 9 2 cannot be j u s t i f i e d at the present stage of the a n a l y s i s , i t w i l l be shown in Appendix III that + 6^ v i o l a t e s uniform v a l i d i t y in T unless 9 2 = 0- t n e same condit ion a p p l i e s , in f a c t , to Qy 9 ^ , and 9^ (which i s the l i m i t of the analys is i n Appendix I I I ) , and i t appears l i k e l y that 9^ = 0 for a l l i > 0. Af ter some manipulation, the one remaining secular term i n Vg can be obtained in the form « iv when Y 2 i s constant (by the argument of page 4.14), and i f i s a lso constant, then the term -e [3ta.j] i n Equation 4.44 i s e l iminated by o i 3 = Z j ^ O - f O ^ . (4.22) At present we have only assumed that 0 2 and are constant. Both these assumptions are , however, v e r i f i e d in Appendix I I I , and f o r Ug and Vg we then have Page A3.24, Equation A3.83. 100 4.21 - 2_ + 8? c o s t + « tos H 4 6 4 »4 (4-33) 31 s«t + rs sc^3b 37- 32.' 2-1 b where the secular term i n v g ( t ) i s el iminated by the correct choice of a. This process of a n a l y s i s , which i s continued i n Appendix 111: leads to the fol lowing r e s u l t s : (*•«*) ^5 * 3 r 31 -t- Z%£ 26 & 4 f*,) . ( 4 . * t ) 6 4 0 + ^ , ) ^ ( 4 . S ? ) S ? 6 6 4 I I. 141 + 135 p-, 3 z ( l + . p l > J O (4.88) (4.89) ( 4 . 1 . ) (4.12.) u^(t) = f*,('+^,)i (i-j*0 -9- +• M t o j t - i_5 cos 2-t £> 6 (4-«3> 101 4.22 ?> 17. 2. (4.14) 2 +• 22-5" p. i 1 i 7 G o 2 t + 7 Co5 A-t v 1 0 f t ) = ^,(i- / O | -T 4 + ^ /"*• 7 Jt'*"t ~ •( + x % s t^i ^ •f 15' + 6 7 S ? t 1 I 64- y.xCj+w,) > j Six. 2.b -63 si^i+b - 3 ^ t - s ^ . k ) - < e S / * , ( l - j O ^ ( l - c a t " } (. - f O 16 C.\.+.i*, ) > (A-.9fc) where the development i s terminated at O ( e ^ ) . Note that the secular terms i n Equations 4.94 and 4.96 are el iminated i n the complete s o l u t i o n for 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 . 3 Y 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 quant i ty . There i s , however, no reason to exclude the condit ion ->- 1 , since the equations of motion are v a l i d for 0 < ^ < 1. In t h i s case Equations 4.86, 4.88, 4.95 and 4.96 contain unbounded terms unless Y q and Y-J are both zero , and i t appears l i k e l y tha t , i n general , Y ( T ) = 0. When Y Q = 0 we can also state from Equation A3.241 that 102 4.23 The values given in 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. I f , however, we make the p laus ib le assumption that ctg, 6g and 6^ are constant, then we may also wri te (from Equation A3.250 with YQ = Y-j = 0) ; S e c 0„ 5 * c e, ( A . too ) In addi t ion to these concrete r e s u l t s , the der ivat ives of a , 3, y with respect to T are at l eas t O(e^), ||- = O(e^) and tP = 0(e°) From Equations 4.85 to 4.100 ( and assuming 8g and 0^ are constant) , solut ions for ^ ( t ) and r ^ t ) can be accumulated that have a constant error O(e^). Secular terms i n the so lut ions for u ^ ( t ) 2 3 and v ^ ( t ) could involve t , t and t expressions (see Appendix I I I , Equations A3.8 and A3 .9 ) , but i f secular terms are el iminated from the 103 4.24 lower-order solut ions no t J expressions w i l l be generated. I t also appears 2 highly l i k e l y that t terms are el iminated i f a ( r ) = constant. With a reasonable degree of confidence we can therefore state that the solut ions for ^ ( t ) a n c ' a r e u n i ^ o r n i ^ y v a l i d for t = 0 ( e ~ ^ ) , but even the most pess imist ic estimate gives a time in terva l of uniform v a l i d i t y 0 ( e " 7 ) . A f t e r some manipulation, these solut ions can be derived as: J. + Cos It - ^ - cos 3 fc & 4 -( 4 . 'oO e ^ / l f ^ ' i M i - ^ ) l o S d - ^ ) c ^ t + 1 1 cos 2.1 ] - £ ID 3 + co^ 4-t IS (^Ci-p,_") sc*.b * J + £ t l f o • Ifed-t-fJ,) J .b - tS" sin. 3b 32-»2-( 4 . I O Z - ) 101}. 4.25 10x4 Ci+^i 7) 3 Z b +• fe3 S c V 4 b + o c v ) , where i t i s assumed, in 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 so lut ion (see Equation 4.78) . . Corresponding resu l t s for £ 3 ( t ) and n 3 ( t ) can r e a d i l y be obtained from Equations 4.101 and 4.102 using Equations 3.5 and 3.6. We then have: + 6 -r 6 >s A < i f c . £ ' - / 0 c o s t /6 (1+ ) ~* U' z? + >35 pi, ? « j t - 2£ cos 3 fc 3 2 - 3 ^ 1 + ^ 1 ) J 4^-d o 2b i o 5 ( l - p ^ " ) est + n co^2 .b ( 4 . 1 0 3 ) * cos4b 3 2 . OCfe") ^ 0 11 sc^zb Si*>.t — _ i £ S C n . 3 b 32. J L 6 4 0 + / X , " ) -+- SH Si«,?-b 12. ] ( 4 . 1 0 4 ) 105 4.26 4.4 A Uniformly V a l i d Solut ion of the Restr ic ted 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 so lut ion 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 fo l lowing form: r L 6 O f t ' ) ( 4 . « o S ) 9 1 / \ ! i± S i * , i t 12. (A-.lofc") where these solut ions 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 configurat ion for the r e s t r i c t e d problem If we select the H, H coordinate system shown in Figure 4 -1 , so that both and m2 are located on the E a x i s , then the transformation equations are H = ^ d.ib -<• cos t ( 4 . | o ? ) ( A - - l o g ) With respect to the ~, H axes, Equations 4.105 and 4.106 can now be wri t ten as: C fc) = . cos b 4-g *4-it 2 107 4.28 - - Sin.t 4 ? 3-4 A 6 if. 8 (4.l«o) In t h i s form the so lut ion 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 in the fo l lowing sec t ion . 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 analys is 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 deta i led discussion of H i l l ' s analysis can be found i n Brown [ 5 3 ] and Hagihara [ 5 4 ] ; for 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 are: 1. the so lar para l lax i s zero .t More s p e c i f i c a l l y the analys is 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 respec t ive ly . I f the ear th 's equatorial radius i s denoted by r * , then the lunar and so lar paral lax TT and T T o are defined by: = arc s i n L a arc s in _r r ' and, since r « a « r ' a_ r ' 7L-C I f the so lar para l lax i s zero, a / r ' i s therefore zero. 108 4.29 2. the solar 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 zero; 4. m x / i s s u f f i c i e n t l y small "to be neglected. • / lA | Note that although H i l l ' s analys is 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 condit ion i s implied by se t t ing the so lar e c c e n t r i c i t y equal to zero. The sun's o r b i t about the ear th ' s mass centre i s consequently c i r c u l a r , so that m-j and maintain a f i x e d con-f i g u r a t i o n independent of the motion of m^. Note also that the Jacobi i n t e g r a l , which H i l l uses to e l iminate 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 zero, 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 U U t ein can ue I C U U L C U uy an as,ym[j lui, i C l i m i t process to give H i l l ' s equations of motion. A s i m i l a r approach also appears i n Szebehely [ l 1 0 ] , but i n the summary given here we f o l l o w Kevorkian i n locat ing the larger mass (m-j) on the negative H axis (Figure 4-2) . The configurat ion 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 selected so that the H * H* coordinate system rotates at uni t angular v e l o c i t y with respect to i n e r t i a l axes, and the earth-sun distance i s u n i t y . We next locate m^ at the o r i g i n of a new system by se t t ing * = - ' 0 - / 0 . . ( 4 . n i > <j = H . (.4.WZ.) ri 109 4.30 Figure 4-2 Primary conf igura t ion , with respect to the H * , H* coordinate system, for 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 wr i t ten as: & - 2.. 4-at + l -r . 3 i - 0-/O + p. X where An asymptotic expansion of the form 110 4.31 C4-.ll?) ( 4 . us) 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 terms and J i appear as 0(y) whereas in p r a c t i c e , f o r a r b i t r a r y u, there exis ts a region i n the neighbourhood of m 2 w i t h i n which these terms dominate the equations of motion. Near the o r i g i n a so lu t ion defined by Equations 4.117 and 4.118 i s consequently not uniformly v a l i d i n the space variables x and y as u -»• 0. This non-uniformity can be avoided i f we take t + « t U.II<O and, i n terms of these new v a r i a b l e s , Equations 4.113 and 4.114 can be wri t ten o(.-3-/3 i - 2. dLka dt ^ s i - Ci->) - ^ r 3 r 3 ( 4 . U 0 ) (4.13.1) where, for convenience of nota t ion , t has been replaced by t . The quant i t ies r^ and now take the form 111 4.32 so tha 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 subst i tute f o r i n Equation 4.120 using Equation 4.123: JLt1 ' 4 (4.12 4-) The values of a and 3 can now be selected from the condit ion that s p e c i f i c terms i n Equation 4.124 are of a p a r t i c u l a r order i n y . I f , for 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 i n the . v i c i n i t y of m^, i s considered in d e t a i l by Kevorkian [55 ] . Con-verse ly , when the planetary g r a v i t a t i o n , C o r i o l i s and centr i fugal accelerat ions are a l l of the same order we have: oi - %fi = dL-p and therefore a = 1/3, 3 = 0 . With these values for a and 3, Equations 4.120 and 4.121 can be wr i t ten as: d-t r , , - , 3 / , I f terms 0(\i%) are neglected and we set (see Equation 4.123 with a = 1/3), Equations 4.125 and 4.126 are i d e n t i c a l to H i l l ' s equations of motion [ 5 2 2 ] . In the terminology of asymptotic expansion theory, H i l l ' s equations are therefore obtained from Equations 4.113 and 4.114 for the planar r e s t r i c t e d problem by holding the var iables 113 4.34 f ixed as u -> 0. The p a r t i c u l a r symmetric per iodic so lut ion of Equations 4.125 and 4.126 corresponding to the i n i t i a l conditions 5 / 0 ) > o ; ^ C o ) - o j 5 ^ ° ^ = 0 defines H i l l ' s v a r i a t i o n o r b i t [ 52 3 ] . This so lut ion 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 v a r i a t i o n o r b i t From the general ser ies s o l u t i o n , H i l l computed a per iodic lunar o r b i t assuming the fo l lowing sidereal periods [ l 1 1 ] : T L = 27.321 661 T $ = 365.256 371 . Here T^ and T<. denote the lunar and so lar 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 for 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 in current use: Sidereal mean motion of moon K (1900) i n radians/sec . E.T. [ 5 6 ] 2.6616 99489 x 10"D Sidereal mean period of moon (1900) .in ephemeris days 27.321 66140 Sidereal year 1900 i n ephemeris days [45 ] 365.256 36556 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 [421*] i n a form compatible with Equations 4.109 and 4.110, i s given below. X = o. <Ho4o s?-&og cost • 4 o. ooooo SS?$8 COS. St •*• o. ooooo ooioo cos ?t + o. ooooo 00007. CDS 'J fc C*4. I Z 7 ) Y =- i , C O 7 7 8 i s o - i l sc^t •» o. oo 151 4i 1 3 sc«. 3b + o. oo ooo 56 7 09 S"fc •*• o . o o o o o 06300 s^7b 4 0 * ooooo 00007. s;«. 1 1 ( 4 . i a e ) I t i s assumed, for 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 larger mass m^  i s located on the p o s i t i v e X a x i s . Equations 4.109 and 4.110 were derived f o r the same i n i t i a l conf igurat ion but with m^  on the negative H axis (see Figure 4-1) . The two o r b i t s should therefore be equivalent i f we set E = - X , H = -Y so that Equations 4.109 and 4.110 take the form: 115 4.36 X(b) = cost 1 - 65 4 6 + cos 3fc 3 .+ 7 6*p,(i+n,Y 16 « 4-8 *4 6 9 1 Now from Eauation 3.21 where fi = T^/T^. Consequently, for the s idereal periods used by H i l l : 9. = 7.4801 32633 x 10 -2 The earth-sun distance can be obtained by expressing the a . u . i n dimension-free form. From Ash, Shapiro and Smith [57 ] : 1 a . u . = 4.9900 4785 x 10 2 L ight seconds, and from Clemence [56 ] : 116 4.37 Mean earth-moon distance q i n metres A = 0.384400 x 10 Ve loc i ty of l i g h t in q metres/second c = 0.2997925 x.10 . We then have the fo l lowing numerical values for r-j and u-|: r 1 = 3.8917 24558 x 10 2 y ] = 3.2979 46189 x 10 5 , and from Equation 3.28 for 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 - 7 . 5 7 6 8 6 * 3 8 * 1 ^ l o " 3 - l . 6 l l S 5 8 6 2 ? % \0~* ( 4 . 1 3 O 16 S L - . P - V W * . . l o " 3 •+• 3 . 6 6 Z I 3 7 3 & S * 1 0 " * 1 + 5". 7117 8ZS6? y. l o " 3 + l . M i S S " € 6 S 8 * l o - 3 ( V ( 3 3 ) 117 4.38 x c t ) = o. <?<) O ? V i ^ g l c o 5 t + O C V ' ) ( 4 . •+.O.. o o . l A - l S 3 i v 7 e _ s ; ~ - . 3 t + 0 ( t " ) (4.(35) A comparison of H i l l ' s so lut ion (Equations 4.127 and 4.128) with Equations 4.134 and 4.135 shows reasonable agreement. Note that the con-vergence of ind iv idua l c o e f f i c i e n t s i s determined approximately by the 3 Vz 3 'A quantity e u-j ; in the present case e = 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 so lut ion I t i s evident from the general behaviour of Equations 4.105 and 4.106 that the next contr ibut ions w i l l be 0(e ), and although a 12 complete determination of 0(e ) terms i s not f e a s i b l e , the functions u-j 2 (t) and v- j 2 ( t ) can be evaluated i f we assume = o • In t h i s case Equations A3.206 and A3.207 can be w r i t t e n : 118 4.39 766 - 4 £ ^ f l - j O 16 4 (Vl3<0 175 ccs 3 b 67J- p ^ O - j O cos 4fc 17-8 32. - s.>-2.b 6 4. 12.9 32-115 / * | C " ' - f O scw$b 12-8 where we have taken YQ = Y-J = Y 2 =. 0 . From Equations A 3 .8 to A 3 . 1 1 , the corresponding solut ions for u ^ ( t ) and v ^ ( t ) are: 23-6? + (OS «>5b 768 48 4e<r?2-b + j - 5 c-n? 4 b 2-^ 6 - 13-7- -t-. ?£_1 COJS t - 137 c o s Z b I % (U. OS) - 4 / + MS cos b + JZ4S c c j 3 t + 4 4 ! <^ s S " t ... 3 x i<*-4 This s i m p l i f i c a t i o n i s discussed on page 4.22. < 1 1 9 4.40 5 4-S 2-^6 «3L? t - 5 i s s ^ b -t- ^  sivv^-t 4- .1 ? x £>4 ^4 - 3 0 7 2 . %S(>o ( 4 . 13*1) where the secular terms i n Equation 4.139 are el iminated when s a t i s f i e s Equation 4.88. The contr ibut ions to Equations 4.103 and 4.104 for £ 3 ( t ) and n-jCt) therefore take the form: - e. \$5\ L ?-3°4 + 105 coot -t 4 - c c o 2 - t + c o s 4 t 4 S ;z$& Z f J cost 1 - 137 &o 2. b 18 M5 C e o t + 3-4 S cos 3 t •+ 4 4 1 cos *t 11-8 ie>2-^- 3 t > 7 l . (4.1*0) 120 4.41 '1 6 fx r> |_o w b -t- 3 £ s^Z-t - _ 2 £ s^ij-b i 4-g1 + e S j 8 5in.b 1 7?-.2-5 sc-vb + 7 7 o sc.v3b -t- 3'*) °i Sivv-5t ( 4 . I 4 l ) Note, however, that these expressions are incomplete because the terms fxf>^ Co$t and - 1 fxf>^. n a v e D e e n omitted from and respect ively (see Equations 4.25 and 4.26). The e f fec t of t h i s omission i s purely a matter for speculat ion, but from previous solut ion behaviour i t seems probable that constant and cost terms should be removed from A r , and s i n t terms eliminated from A t I f , therefore , these terms are neglected, then applying the l i m i t process (1 - l ^ ) 0 to and we obta in : V b ) - .1 . + b CA.14-2.) Co5 2 b ^ 5 e ' 7 - ft, 1" c o s 4 h . 2 S 6 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 in cost and s i n t involve the funct ion (1 -^V» and as ( l - j ^ ) 0 these w i l l a lso be e l iminated. 121 4.42 Cfc) = - s>v^ t 2.56 ( 4 . 1 4 3 . ) Equations 4.109 and 4.110, def ining motion with respect to the E, H axes, can now be revised as shown below. z : c t ) = -cash 48 *4- [_ 144- S - cc-s 31 L 4 -S I t J J SIS £ "-p,1" cos 5"t 2-St o a r ) ( 4 . H 3 C t ) = - S<«.t l*S Ur I 144 • ? t 3 /6 <S (. 4-S <?4 £ 5 G , l ^ y Jin. S t 2-^ (4.14?) 122 4.43 I f we evaluate the addi t iona l , c o e f f i c i e n t s using the numerical values for e and y-j given on page 4 .37 , then | 114-77 _ I-*? p,Z ~l = 2 . 4 / 7 0 8 32-0 IS *. \0~** ( . 4 . 1 4 6 ) L 1 4 4 1 6 3 L 42 <?6 ~> ^£ e1^^,"1 = .3.0572 7?fc(?7 * l O _ f c . t. ...(**. I 4 « ) i n which case Equations 4.134 and 4.135 for X(t) and Y(t) can be revised as : X(fc) = O . <no4<3 . 441S7 cost t o . O o ! 54 ^ 3 4-1P cos3t + o . ooooo 3e>S72 t c . s S t + o a ' 5 ) C4I49) Yet) =• I . oo765 0 4 3 0 5 si-t -+ o. 0 0 1 S 4 S 3 4 T O s^3t + 0 . ooooo 3e>5 72. 5i«-?t.. 123 4. This so lut ion shows a c loser 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). I t must, however, be emphasized that Equations 4.149 and 4.150 are based on an incomplete analysis and that the conjecture applied i n der iv ing Equations 4.142 and 4.143 cannot be substantiated. The resu l t s of t h i s sect ion should, i n consequence, be regarded as informed estimates. 4.5 Discussion In contrast to H i l l ' s approach, which appl ies only to the r e s t r i c t e d problem, the analysis of three-body motion presented here i s v a l i d for any system s a t i s f y i n g the condit ions out l ined i n Section 4.1 At l eas t i n p r i n c i p l e , using th 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 so lut ion i s no longer v a l i d i n t h i s case because the Jacobi integra 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 zero, so that some inaccuracy might well be ant ic ipated when Equations 4.127 and 4.128 are used to describe lunar motion. We can, i n f a c t , estimate the error introduced, by comparing Equations 4.109 and 4.110 for the r e s t r i c t e d problem with a corres-ponding so lut ion 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 9 i s determined by: 121+ 4.45 - 1. ,- e' tost + e 6p, .J. .+ c&si-b 6 * c n - p , y * . J Sirvb (4. IS-*) I f 5, H axes are chosen so that the o r i g i n i s located at and m-j l i e s on the negative E axis (as in Figure 4-1) , then from Equations 4.107 and 4.108 we obta in : . 3 . C O = TTT7" 7T- L J ( 4 - 1 * 3 ) - £ 5 1..32CI.4- p,)s- J c ^ x t 3 . e6- n,.**s 3 t + o( £*) L ^ '*;<,(•-/O L $2- C H - H . ) i J ( 4 . 154) i t 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 2 /m 3 = 81.3024 + and therefore : (1 - u 2 ) = 1.2150 31396 x 10" 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 . Z 8 4 0 ^6S"<?6 x i o ~ 6 , ( 4 - . i 5 6 ) Neglecting the mass of m^ w i l l therefore introduce errors of approximately 4.3788 x 10" 6 and 1.0947 x 10" 6 in to the 5 and H components respec t ive ly . S imi lar errors can be ant i c ipated in 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 for a number of astronomical con-s tants , 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 that a comparison of H i l l ' s o r b i t with resu l t s derived from the present uniformly v a l i d so lut ion may be mislead-i n g . 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 t t] i s , for example, neither necessary in t h i s analys is nor , i n f a c t , evident in Equation 4.153. A deta i led r e v i s i o n of H i l l ' s approach can be found in 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 solar distance r-| -* °°; 2. the so lar 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 zero; 4. the mass r a t i o + " 3 + 0. M ( + M . ^ + r v v j Their analysis consequently takes into consideration the combined mass of the earth and moon. Jacobi ' s in tegra l i s s t i l l , however, appl ied to reduce the equations of motion, so that i n e f fec t the lunar mass must be n e g l i g i b l e for the r e s u l t s to be v a l i d . The analysis of Section 4.4 indicates that the convergence of Equations 4.101 to 4.104 depends, to a s i g n i f i c a n t extent , on the quanti ty e^u-| , where j-i, and £ = y •/ We should therefore expect p a r t i c u l a r l y convergent so lut ions for 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 apply, for example, when two stars forming a close binary o r b i t a comparatively d i s tant s t a r , the masses being of s i m i l a r magnitude. Note, from page 2.26, that provided TX- « 1 we are j u s t i f i e d i n assuming ^ r l two body o r b i t s f o r m-| and the mass-centre of m9 and m v This s p e c i f i c 127 4,48 case w i l l be given more deta i led consideration i n Chapter 6. Next, how-ever, we apply these general three-body resul ts to the four-body problem. 128 5.1 5. THE FOUR-BODY PROBLEM 5.1 Introduction Now that a uniformly v a l i d so lu t ion for the motion of p-j, p 2 and p 3 has been der ived, we can proceed to invest igate the motion of p^ as out l ined in 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 equi l ibr ium) s o l u t i o n of the two body problem. S i m i l a r l y , the analysis of four-body motion takes as i t s foundation equi l ibr ium solutions of the r e s t r i c t e d problem. A deta i led treatment of the r e s t r i c t e d problem can be found i n Szebehely's author i ta t ive text [ 1 ] , and only those aspects which are necessary for the subsequent development are described here. Although the method of so lut ion i s b a s i c a l l y i d e n t i c a l to that adopted for the three-body case, the analys is 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 so lut ion for a four-body o r b i t i s to have an error ©(e 1), then i n p r i n c i p l e the motion of p-j, p 2 and p^ should be known to an accuracy 0(e ) . In the present case t h i s four-body so lut ion i s consequently o l i m i t e d to an error 0(e ) by the resul t s of Chapter 4. The analys is should therefore be regarded, not as exhaustive, but as providing the basis for fur ther 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 analysis as f a r as error terms 0(e '*) to define a uniformly v a l i d so lu t ion w i t h i n an error 0 (e " ). A s i m i l a r l i m i -ta t ion a f fec ts the present d e r i v a t i o n . •129 5.2 5.2 The Restr ic ted Three-bo.dy-Problem In Section 3.3 we considered the process by which the four-body problem could be reduced to a r e s t r i c t e d problem involv ing p^, Pg and p^. The motion of p^ in the r e s t r i c t e d problem i s described by the fo l lowing equations: (Equations 3.25 and 3.26). Equations 5.1 and 5.2 possess f i v e homo-graphic solut ions 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 feas ib le to invest igate motion in 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 wri t ten as 1 -Mr. ~ h i t : - o 130 5.3 1 = o C5.G) since = n.^  = 0 (from Equations 3.38 and 3.39). To cast Equations 5.5 and 5.6 into a t ractable form we require some method of expanding the nonlinear functions _J as ser ies in x and y . I t i s , however, important that the f i r s t few terms of these expansions should represent the nonlinear functions accurately wi th in a region containing the complete s o l u t i o n . For the case of perturbed two-body motion considered in 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 , for 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 that motion i s not, i n general , confined to such a small region. We consequently require an expansion which provides , with r e l a t i v e l y few terms, an accurate representation ins ide a s i g n i f i c a n t l y larger region. The nonlinear functions can be rearranged as: 4-I + where l r L - j l denotes the distance between p^ and a Lagrange p o i n t . Now so that 131 5.4 Cs.l) Equation 5.7 can therefore be wr i t ten in the fo l lowing form: I f we set (5.|o) *> _ L r i + ^ c ^ b ) then a general series expansion of the funct ion [1 + p(x ,y) ] in te rva l - a 6 p ^  g can be wr i t ten as •3/4 on an Cs". i i ) In pract ice the summation i s terminated at some f i n i t e value of i , leaving a residual error which i s , hopeful ly , smal l . The constant c o e f f i c i e n t s k . w i l l then, in 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 in Equation 5.12 i s s u f f i c i e n t . The foregoing discussion appl ies to motion which takes place ins ide a region of f i n i t e s i z e . As indicated on page 5 .1 , however, solut ions of the l i n e a r i z e d r e s t r i c t e d problem are central to the analys is of four-body motion. Consider, therefore , the e f fec 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 2 x , xy, y and higher powers of x and y can be neglected. We then have, 132 5.5 for Equation 5.10: i ^ 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 der ived , using the expression given i n Equation 5.15 to s i m p l i f y the nonlinear funct ions . Af ter some rearrangement the l i n e a r i z e d equations of motion can be wr i t ten — 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 Equations 5.16 and 5.17 take the fo l lowing form: 5.7 = CK-O r ^ 3 I • t 3 1 Motion within an i n f i n i t e s i m a l l y small region about a Lagrange equi l ibr ium point i s now defined by the so lut ion of these equations. Note that kg = 1 for 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 ] (/. at) The homogeneous so lut ion of Equations 5.23 and 5.24 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 so lut ion depends on the p a r t i c u l a r Lagrange point to which i t re la tes . There are , 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 separately. ( i ) At the col l i n e a r Lagrange points the homogeneous so lut ion of the v a r i a t i o n a l equations can be expressed as :<-b) A ; e Cs.zf) B: e Mb C5.28) 135 where, in general , there w i l l always be one unbounded term [1 1 k ] . ( i i ) Close to the equi la te ra l t r i a n g l e points the homogeneous s o l u t i o n i s bounded when 5.8 < r 2- - J - - 4-(k 0+k,) 3 k * (see Appendix IV, Section A4.1) ( i i i ) When V-3 _\_ — -X - . 4-Ckp + O 3 k.1 C 5 - . 3 0 ) the v a r i a t i o n a l so lut ion about and contains secular terms of the form t s i n t , tcost which, i n general , render the so lut ion unbounded [ l 1 5 ] . ( iv) I f h i 17- 3 k (S-31) the so lut ion near the equi la te ra l t r i a n g l e points and Lj. contains terms of the form e ±^ tcos(cot + 8) so that , i n general , the so lut ion 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 solut ions indicates that motion cannot, in 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 . Bounded motion may however, s t i l l be possible in cer ta in instances when nonlinear 136 5.9 e f fects are taken into consideration [ I 1 7 ] . In the fo l lowing analysis of four-body o r b i t s we consider only those values of y 3 for which Equation 5.29 i s s a t i s f i e d . As e -»• 0 the equations descr ibing four-body motion reduce to corresponding equations for the r e s t r i c t e d problem near and L ^ . We sha l l consequently continue to use and Lg as points of reference, although i t should be noted t that i n the four-body case no posi t ions of equi l ibr ium a c t u a l l y e x i s t . 5.3 Four-body Motion Near 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. This corresponds to case 9 i n Table 3-1 ( in Section 3 .6 .2 , pages 3.21 to 3.26). Equations def in ing the motion of p^ were derived i n Section 3.4 (Equations 3.33 and 3.34) , and for convenience these are re-s tated below. Posi t ions of equi l ibr ium at s p e c i f i c time instants can be defined by equating der ivat ive terms in the equations of motion to zero; a discussion of th i s topic and i t s impl ica t ions i s given in Appendix I I , q . v . 137 5.10 S ^ — * ! r - l 3 t- *" I r I * When Equations 5.32 and 5.33 are subst i tuted into these expressions we obta in : 3 138 5.11 .where S ( K , V , b) = f-i cost •+ f., t^.-O (s--3s) Note that , because (? L » n L ) i s a f i xed p o i n t , no der ivat ives of ^ and appear i n Equations 5.36 and 5.37. We can evaluate the expressions S ( u , v , t ) and T ( u , v , t ) from Equations A1.6 and A1.7 i n Appendix I (see pages A1.2 - A1.4) . Next, however, the nonlinear functions _J and - i — 3 must be expanded in terms of e. The posi t ions of „ ~ . ~ J — - ~J J r - , . -u ^ / i i r \ i x. _ / i i n / i u...«. x i . — l , , x . v H 2 u n u [-/^ » y i v c n l i t L ^ u u i> i UII j T • » O i - r • i WT 9 U M U J C v > i u u • O i i o also depend on the small parameter e. This feature complicates the process of expansion considerably, and d e t a i l s of the analysis are therefore relegated to Appendix V( Section A5.1) . Eventually we obtain the fo l lowing r e s u l t s : | k0u..+ k, j \ it. + V / v f J i - 2 . j O J ^ C 5 - . 4 0 ) 139 5.12 - £ O C V ) - 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 for these values i s given on page 5.12 . We now have ^ 1 1 — - 2 V - «<- [ * - ( k 0 + k, > 114.0 5.13 £ ^  V •+ ZU, + I L W ^ ^ C I - 2 / 0 - vy £ I + 3 k , ) j ^ - <f ' l - O + O^ f c 1 " ) , ( 5 . 4 3 ) and, by subs t i tu t ing for £ L and n L i n Equations 5.20 to 5.22, at and these equations can be wr i t ten as: e £ 11 - zv...+ 0,u-+ O ^ v ^  = ^ ( i _ k c V + O C V O 0-44) e £ v •+ zi. Ar o^u. + o^v J = ^ 0 - k 0 ^ o a O . . (5 . 45) In th i s form the l i m i t process e 0 appears to be i n v a l i d , but the constant terms involving kg are , in f a c t , dependent on e, as demonstrated below. When kg f 1 there w i l l be some error i n representing the functions _ L _ and 1 at L- and U (where eu = ev = 0 ) , which i s caused by the process of truncat ion described on page 5.4.. I t i s not unreasonable to assume that t h i s error i s of the same order as the in terva l wi th in 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 1 ) i t fol lows that (k Q - 1) = O f e 1 ) . A proof of t h i s conjecture f o r Chebyshev polynomial approximations i s given i n Appendix VI ( in Section A6.4) . 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 lass of. the truncation error w i l l depend on the leading term i n the 141 5.14 solut ions for eu(t) and e v ( t ) . These solut ions depend, i n t u r n , on quant i t ies 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 s | 2 k ^ . i ^ e^j and e s J2k, p ^ J f j • I f e^ and fg generate no secular terms in the solut ions for eu(t) and e v ( t ) , we are therefore j u s t i f i e d i n assuming ( ) = O(e^) so that , from Equations 5.44 and 5.45, 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 solut ions of Equations 5.44 and 5.45 contain only per iodic terms of the form cos(to st + 0) and cos(coLt + <j)).+ From Equations A4.19, A4.21, A4.28 and A4.29 we have: ko + k, -+ [ -vdc^w,) + k * - ^^c\-fAZ)] *• (s . * o ^ ^ I + k e +k ( - [ 4 ( k 0 + 0 + W,2" - 3 k ( % ( i - / 0 ] V <*.*?0 and consequently, since k Q and k-. are constant: j . . Appendix IV, Section A4 .1 . iij.2 5.15 3A For a binomial expansion of the nonlinear funct ion [1 + p] A kQ - 1 3 and k-j. = y » i n which case ^ - 1 ( S . 5 o ) V ^ ° . ( 5 . 5 0 where the equal i ty condit ion applies when u 2 -»- 1. The change in these 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 pract ice the changes are s u f f i c i e n t l y small that Equations 5.50 and 5.51 can s t i l l be used (see Appendix V I , Section A6.5) . I f no secular terms are to be generated by e 5 ( t ) and f 5 ( t ) , we must therefore require that y 2 f 1•' When th is condi t ion 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 var iables 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 so lut ion 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 2 f 1 i s consequently i m p l i c i t i n t h i s ent i re a n a l y s i s . 11+3 5.16 where these expansions are to be uniformly v a l i d in 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 der ivat ives 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 non-homogeneous p a r t i a l d i f f e r e n t i a l equations where, for the i ' t h power of e : 1 hi?- J Here the functions S and T involve u c to u . , and v c to v . , , but not o l - l 5 i - l u.j and v^. The f i r s t pair in t h i s sequence are it*- *t *" l\ + . + o A u. 5 + o 3 v s = e"5/y• Ci-iO - ^ f t ) t ts.S7) dl7- dk It. and i f the homogeneous solut ions of Equations 5.56 and 5.57 are denoted by u 5 ( t , x ) H and v 5 ( t , x ) H r e s p e c t i v e l y , then from Equations A4.30 and A4.31: 'ui<-*F')fr = o-Ct?) o-s [ fc>st + 6<lv)^ + h(t) cos [ iO ut + e/>Cr)J (5.58) * 5 - o s a c e ) cc.5 [ <o5fc + 9 * t t O ] + fflu U r ) c« ]_ mufc + < £ * ( - & ) ] (5.51) where, from Equations A4.34 and A4.35: &* Co) - 9 C v ) + 2 - 3 y ( s . 6 o ) 11*4 5.17 Homogeneous solutions of higher order can be accumulated into Ug ( t ,x )^ and V g ( t , r ) H i f the a r b i t r a r y functions a ( 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 for a comparable process in the three-body case) . A l s o , by r e - l a b e l l i n g x as e t , the non-homogeneous so lut ions for u^ and v.. (where i > 5) can always be considered functions of t , and consequently no der ivat ives of these quant i t ies with respect to x need appear in the sequence of p a r t i a l d i f f e r e n t i a l equations. Within an error O ( e ^ ) , 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 & l + O t V O Is b T t V , v , f c ) = \ T. C t ) £ : + O C V ' ) 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 wri t ten i n form given below. d\s + 2_5us + 0 z u . 5 t O j V s «. e~Srf(i-k0} - %\Y^"I e s C t ) cJt1- 5b • L 5t l 5b 5 Ct) - 2- - 5v <3tatr __5 OC.) T 6 C O - Z k f ^ e / t } 5 t5V 3V - 7-• i t * 5b 5.19 x 5 " ? Tf C O - a i c , ^ ^ t ? ? f t ) - 2- a V 5 L a t a r • a v 'oC.) at 1--a t •4- J, a ? + oly9 - C O -- 2 a V - - 2>J$ _ + T V J - ^ \ oCt) I t 1 - a t J + z>a g + o x a ^ + 0 3 v g - T C O - Z W , ^ e s ( 0 at a t 2- -OCe v) i - ^Ct+jO' 1-06O a , -v p V5 -t- ciu.5 OCi) _^«-<} ...-.2 a_Vq + 0 ^ + O^V^ i f a fc s / t ) - ^ . ^ A ^ C O -I r ^ . aTai atr a v v . - ayt + z a x u s - av s ] at a t a r a * J ( h\ 4 aaa, + 0 ^ a, + = T ^ C O - ^ e</0 at1- at u ( ? ? ? ) 2- ^"Vj au.5 - a%/ 5" 0 ( £ O it;1-1 j + ^.5 7 -f -»- aj*t + 2 a \ s ^w.^  ? 1 ae- at J o 6 e ^ L at1- at ik^V a v j 114-7 5.20 c>fc - 2- a \ i 5 -1 • — * \ a r a > 1 bv- at 1 0 c , ^ L it*- bt ^ t a t r bv J OCe~)> L bt^ b t at*- J o C l> , u 5 e 5 (3k, + k^) 1- V 5^ s Ck,+3k,.) o(0 at - 7-oCe*) at* aw- ^ J o o C e 3 ) o C O 1i+8 5.21 J 0 C O As before (page 4.8) the subscript 0(e 1 ) denotes terms of order e 1 i n the associated asymptotic expansions. -5.4 Uniformly V a l i d Solutions The method of so lu t ion 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 for the three-body problem, where a uniformly v a l i d so lut ion i s determined by e l iminat ing a l l secular or unbounded terms. Because the analysis 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 ent i re development i s given i n Appendix V, Section A5.2 . The non-homogeneous functions i n Equations 5.68 to 5.79 can, i n general , be wri t ten i n the fo l lowing form: • —I r '4 J ( 5 . 8c») / 1 11+9 5.22 -where E and F are defined i n Appendix IV, Equations A4.J36 and A4..37. Corresponding non-homogeneous solut ions are given in Section A 4 . 2 , and these furn ish the required condit ions for the e l iminat ion of secular terms. For Equations 5.68 and 5.69, and a f te r subs t i tu t in g for e,-(t) and f K ( t ) from Equations 4.101 and 4.102, we have: A" ( I + |A, ) i ' F* Cb,X) = fe" ^ C\- k D) + ^ K l f i f t d - ^ • (sVs'i) 8 O + | 0 A i u C i HlJ III oi i * »v- V-^ MU ^ « W I I »^ ***.v- . _ _ . < v . . . . > "I " ' w * " ' • * ' equations A4.40 and A4.41, the corresponding non-homogeneous solut ions u 5 ( t ) N H and v 5 ( t ) N H can be obtained from Equations A4.44 and A4.45 as: 150 5.23 -S L *^.5 Si.«vt 4- O^Qob (S- .84) ' 5 O - O { { o u C u>5* - t J ^ ) C o J - - I ) ) c o S t In these solutions for U g ( t ) ^ and v ^ ( t ) N H terms associated with can be obtained d i r e c t l y from the corresponding short period (cos) terms 2 2 i f u s i s replaced by (note, however, that (u) s + co )^ becomes 2 2 - (u> s - wL) a f te r the interchange). A s i m i l a r instance of t h i s type of symmetry i s mentioned in Appendix IV, page A4.13. We next consider non-homogeneous solut ions of Equations 5.70 and 5.71 f o r Ug ( t , r ) and v g ( t , T ) . The functions S g ( t ) and T g ( t ) can 151 5.24 be evaluated from Equations A1.6 and A1.7 , and from Equations 4.101 and 4.102: e 6 C b ) *sb (b) * p., C i - j O £ n. s» z t I t therefore fol lows that hilt-- Sin.Zb ( 5 . S 8 > { 1^1 - S:~2.b + toZb The expansion of der iva t ive terms using Equations 5.62 to 5.65 i s a straightforward but lengthy operat ion, and for t h i s reason i s relegated to Appendix V (Section A5 .2 ) . When the der iva t ive expressions i n Equations 5.70 and 5.71 are accumulated using Equations A5.50 to A5.70 we obta in : 152 5.25 5 t r ' ire L a t av J ar av 3 I ar a r J t t (s\ qo) 2-- . 2 . J 5a 0 Qo C w s b -f av ae>0 at av and at; a r (see page A5.15). 153 5.26 (5.1') Secular terms i n the non-homogeneous solut ions f o r Ug and Vg w i l l be generated by a l l the expressions in Equations 5.90 and 5.91 that involve p a r t i a l der ivat ives with respect to x. I f , for 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 w r i t t e n : 2 co. + COS (JLt +• COS O j t 151+ 5.27 •+• cos c O u t (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, therefore , to consider short -per iod terms in the fo l lowing a n a l y s i s , as corresponding resul t s f o r the long-period terms can be obtained by inspect ion . I f we now take av av J L av av J I atr -S L atr av s 6 - z o 5 o>5 f c ^ 0 / - a p a £ o , ^ e / ? •+ 2-5" J ^ s ^ B , + <feo9c>7 L av yc J L .a^ J 5, = | ^ 0 s^0o* + A o39„ <*>80*? - 2.5" ^o. L av av j t at av a r then, from Equations A4.50 and A4.51, secular terms i n v o l v i n g u>s'are el iminated from u g and v g when: 155 5.28 After subs t i tu t ing for YQ» Y-J » <$0 and 6-j , Equations 5.98 to 101 can be expressed as: +- a, = o = o • = o (if. io«.) ftptoC ^ Z . ^ ^ ^ " 1 - 0 - 4 ^ 0 ^ + «9tt^$0^ 2 ( « 5 l - 0 - ^ s ' j Now from Equation A4.34: - o (£. las') 156 5.29 0 6 T arctcxiv- f 2.60 so that we have si* 0 = sc ,^ 9 to* 4 4 ^ ) ^ -+ COS 9 $ ZcJ5 Cos B = cos 9 S in. fii 2c0 s' ( 5 . 'o?) A f t e r subs t i tu t ing for the corresponding values of sinGg and cos6g Equations 5.102 to 5.104 can be wr i t t en i n the fo l lowing form: 2 - O : i «-o 1 2 C 3 C L 0 0o 2_ where c-|, C2> c^ and c^ are constant quant i t ies i n v o l v i n g Q s , to s , , V 2 and v-,. 157 The only possible so lut ion of these equations i s 5.30 J L ( Q-o ") = O (5.117-") and consequently we obtain A l s o , from the symmetry of Equations 5.92 and 5.93: A l l the der iva t ive terms i n Equations 5.90 and 5.91 for E*( t ,x) and F* ( t ,x ) are therefore e l iminated, so that we have 158 5.31 F*£lO -+ Cos Zb The non-homogeneous solut ions u g ( t ) and v g ( t ) can now be derived from Equations A4.40 to A4.45 ( in Appendix IV, Section A4.2.1) as: a lb) V> S - 1°*-^ KM«-JO+J. ^ - j * , cos cJs t r 159 5.32 + jKt Cos e«?i.b + ' ( [«oJ-+ 4--o 3]- + I" 1 ] 3_ 2. V 6 C f c ) = - ^ 4 px si-_ tos b - cos o s k 160 5.33 4 f*, Si*.Zt . ( ^ - 4 X ^ - 0 - p j cos Zt The fo l lowing resu 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 solut ions for Equations 5.78 and 5.79. air at air av ^ 3 av = 4 k = O a_p3 = constant 161 5.34 = constant ( 5 . 1 2 7 ) = constant CS.\T3) = constant . ( s . a f ) 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 case the solut ions for form given below: A6.5), and in t h i s Ug(t) and v,-(t) reduce to the (5.134) 162 5.35 The s i g n i f i c a n t reduction in U g ( t ) ^ and v ^ t ) ^ (Equations 5.84 and 5.85) resul 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: C 4 o 9 0 - C i O , Sl«.0 D 7 ( f f S S ) t^s"1 - O The values of a . and b. depend on ^ J & U i and ua. at? at respect ive ly (see page 4.20 for an equivalent d i f f i c u l t y i n the three-body case) , which prevent the rigorous determination of a-j, a^, b-j, b^, 6-| > 02' *h a n c ' ^2" However, from the general trend displayed 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 zero. When t h i s condit ion i s s a t i s f i e d : a, I P 5 c^eo - iOs A s 9 e ^ ( ^ 1 3 ? ) 163 5.36 ( * . « 5 l ) 8, - L - ( ] 9, ,4 t 0 5 A s s C v v © t 1 142.) where: r s 143) 161+ 5.37 + 6 2, ( S . l S o ) *3 (s.isz) and ( i = 1,2, ••• 4). corresponds to when w s i s replaced by i n Equations 5.149 to 5.152. 165 Considering only the short-period terms i n u g and v^ , from Appendix V, Section A5.2 we. have: H 5 5 which can be expanded as: 5.38 CoS (5 .1 S3 ) H 3 When 30j and d j 1 ^ "are zero, from Equations A5.245 and A5.246 a v h-e (Appendix V, page A5.69) both - aa 9? j and ja .Q, + a o e j _ J 166 5.39 are zero, so that terms 0(e ) can be el iminated from Equations 5.153 and 5.154. • Af ter subst i tut ing f o r a^ and a Q8^ from Equations 5.135, 5.137 and 5.141, u 5 ( t ) H and v 5 ( t ) H can be wr i t ten in the form: + oC*fc*) .... (s.i so When the homogeneous and non-homogeneous solut ions are combined, the terms aQCOs(wst + 6 Q ) , Q saQCOs(w st + 9^*) and corresponding long 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 s i m i l a r cance l la t ion might be ant ic ipated between components of the solut ions for u g ( t ) , Vg(t) and corresponding" 0(e) terms i n Equations 5.155 and 5.156. The functions u g ( t ) and v g ( t ) are defined in Equations 5.120 and 5.121, but i n more compact notat ion : 1 6 ? 5.40 5 * u 0 * -2- u • 2 . J , i ^ « i . . / i . N _ v 6 C t ) *JUC « s l - co^) 168 5.41 Although a proof i s not given here, the cance l la t ion a f fec ts 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 pos i t ion to accumulate, are given below. (sr. 151) l* k' /7 t./*</ /^'-/0 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 wi th in a constant error 0(e ) , where the so lu t ion i s uniformly v a l i d for 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 Lg , so that we may consider each equi la tera l t r i a n g l e point as the l i m i t i n g member of a family of four-body o r b i t s . I t should be emphasized that these unique solut ions do not account for any perturbations other than those r e s u l t i n g , from the inf luence of p-j. The question of s t a b i l i t y cannot, for example, be resolved without an analys is 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 family of o r b i t s depending on the parameter 6 and provide the generating o r b i t s for Case 10. We may deduce from the general Section 3 .6 .2 , q .v . 170 5.4 character of Equations 5.159 and 5.160 that per iodic 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 . ' - r v S C ^ - 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 inf luence on so lut ion behaviour. 171 6.1 6. SPECIFIC SOLUTIONS 6.1 Introduction The solutions derived i n Chapters 4 and 5 for three and four -body motion permit, wi th in the l i m i t a t i o n s of the primary model, a com-p l e t e l y general choice of values for , y,, and r ^ , the distance between p^ and the mass-centre of a n d Three p a r t i c u l a r cases are considered here, two of which re la te to three and four-body motion for the sun-earth-moon configurat ion of our solar 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 predicted, and applies 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 period. 6.2 Three-body Orbits The solutions 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 i n the fo l lowing form: £ (b) = Ii- y.^ - fe1, p / i - f O - pJ^Vx.)1, 2. + l*\ ~l Cos t + . 10S e^pt, ( I- j / O l?S ( I + p, ) 172 6.2 -cc.s2t £4-• JL 6 "VI ('-/-O cos k'e ( o . l ) € ( H-J* r- Sin. 2-fc S 12. L 33- I0Z40+ / O J J 3^ 63 e ' 0 -^z.") r:*u4t 256 .( .t .O I f ( t ) = : p3 - Pa. + £ 6 * / z ( | + M i ) . cos b ' ^ V ^ i 0 - f O L 5 z 3 z d \ + f O ^ 173 6.3 + cos 2 k 64- 32. (6.3) 8 12-L 3z 10x4.(14-^, ) J J 32. 2.5-6 We need only consider the o r b i t of one p a r t i c l e , ' and in t h i s sect ion at tent ion i s therefore confined to the motion of p 2 -Equations 6.1 and 6.2 descr ibe , with an accuracy O(e^), an o r b i t which i s periodic i n t = 2TT. With t h i s as an i n i t i a l estimate, ' c f . Equations 3.5 and 3.6. 6.4 .we .should be .able to f i n d periodic solut ions of the o r i g i n a l equations of motion (Equations 3.31 and 3.32) using a numerical approach. I t i s important, however, that the computed o r b i t be per iodic in t = 2TT and not t = 2TT -i- 6, where 6 i s some small quant i ty . One method s a t i s f y i n g th i s c r i t e r i o n has 'been out l ined by Bennett and Palmore [59 ] , for which the theoret ica l basis i s given below. 6.2.1 Periodic solutions of y ( t ) = B(t)y + f ( t ) I f X(t) i s a fundamental matrix so lut ion of the equation x = A(t)X (6.5) A X '(t) = - X~'(t) fl(t) ( 6 . 6 ) Now suppose where Y(t) i s a fundamental matrix so lut ion of U.8) 175 6.5 We then have, from Equation 6.6: j u. L -* t i t ' and therefore , i f t Q = 0: Y~]Cb)y(b)= j 1 ~ ' L X ) $ C Z ) A X + constant. ( 6 . i o ) 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 no per iodic so lut ion of period T except the t r i v i a l s o l u t i o n y = 0. In t h i s case i f f ( t ) i s T - p e r i o d i c , ^ there e x i s t s a unique T-per iodic s o l u t i o n of Equation 6.7 [ 6 0 2 ] . I f Equation 6.11 i s to be T - p e r i o d i c , then: T y c T ) 4 ( o ' ) -f f y c T ) y~lc-v) icx) AX = ^ to') c&.>o + T h i s contract ion i s used to denote that a funct ion i s per iodic with a period t = T. , 176 6.6 which defines y(0) as: - i r Co) = [ y - ' C r ) - l ] y.-'CuKctOAV . . c&.l3) The T-per iodic so lut ion can then be wr i t ten in the form t-t-T Now suppose that p(t) i s the so lut ion of Equation 6.7 s a t i s f y i n g the i n i t i a l condit ion p(0) = 0. In t h i s case [61 ] : t p ( t ) - j " Y C t ) W r ) K r ) ditr < 6 . » 0 and therefore , from Equation 6.11: tjCt) = YCtOyCo) + j=(t) . ' (6.ifc) I f Equation 6.16 i s to be T - p e r i o d i c : 177 6.7 YCT) 3 CO ) + pCT) = yCo*) Cfe-i?) so that 3co) =• - [ v c r ) - x ] pCT) . a . i s ) We now have s u f f i c i e n t information to generate a T-per iodic . so lut ion i t e r a t i v e l y . The algorithm given i n the next sect ion i s 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 condi t ion . The proof of convergence appearing i n [59 ] does, however, apply to both algorithms. 6.2.2 I t e ra t ive method to determine a T-per iodic o r b i t The system equations are wr i t ten i n the general form iCt) = ^ [ xCt") , t] U.n) and we expand the function g [ x ( t ) , t ] about a T-per iodic estimate x k ( t ) to obta in : o-vc 178 6.8 Sett ing x = x ^ + 1 , Equation 6.19 can be expressed as where the terms 0(|| * w + , - * k II*) are neglected. VJe now make the fo l low-ing associat ions between Equations 6.21 and 6.7: H (t) x Ct) • U.ix.) = ^9 [*k. =ck+|.' (6.as) ax fCt) s . - 3 [ = c k . t J " ^ [ * k J t ] * k 0 * 0 2x and require that the so lut ion 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 for the so lut ion estimate x^(t) with zero i n i t i a l condi t ions , and denote the r e s u l t by p ( t ) . 2. Integrate the matrix equation v c t ) « . ^ [ x k l t ] y e t ) C 6 . z s ) ax over the in terva l 0 s= t ^ T with Y(0) = I (the i d e n t i t y matr ix ) . This generates the p r i n c i p a l matrix so lut ion 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 xk +-|(0) from Equation 6.18. 4. Determine x^+-|(t) from Equation 6.16. 5. Set x^(t) = x^- jCt) and i t e r a t e from step 1 u n t i l s a t i s f a c t o r y 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 poss ib le , because of accumulated e r r o r , for the invers ion process to destroy con-vergence. In t h i s case the t h i r d step of the algorithm may need some modif icat ion to avoid inaccurate elements i n the ca lculated vector x k + 1 ( 0 ) . + 6.2.3 Periodic earth o r b i t For the sun-earth-moon conf igurat ion of our so lar system the constant ft i s determined by the r a t i o of the s idereal mean motions of the sun and moon as: ft = 7.4801 32855 x 10~ 2 . Note the s l i g h t di f ference between t h i s r e s u l t and that quoted for ft on page 4.36 which ar ises from the use of current values for the mean This d i f f i c u l t y was encountered i n der iv ing the. three-body o r b i t s of Sections 6.2.3 and 6.2.4. The two elements adversely affected by the matrix inversion were, i n i t i a l l y , both zero , and convergence was restored by re ta in ing these values i n subsequent i t e r a t i o n s . 180 6.10 motions (given in 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 24558 x 10 2 y 2 = 0.98784 96860 l - y 2 = 1.2150 31396 x 1 0 - 2 The corresponding o r b i t for the earth can now be obtained from Equations 6.1 and 6.2 as: 5 C t "i = 1 . 7 - 1 3 2 < ? 2 3 3 4 *• lo"" 1" +• %.X£iiO 63^34- x lo~'° Ccr^3t 181 6.11 + i . oi~?2-6 v (©"^ s;~zfc • - 1 . f ^ 2 - f&joo s xoX° sin 3fc * 1 - 6 3 0 ? 83o<?4 mo'^siiv 4 t . This so lut ion i s shown in Figure 6-1 in 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 pos i t ion of p^ a^ successive instants of time separated by At = 2IT/100. To v e r i f y t h i s so lut ion the o r i g i n a l equations of three-body motion (Equations 3.31 and 3.32) were solved for 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 instants over the i n t e r v a l 0 ^ t < 2TT the quantity | x k + 1 ( 0 ) - xk(0)|| reduced below 1 0 " 1 3 i n f i v e i t e r a t i o n s , ind ica t ing rapid convergence. A Fourier i n t e r p o l a t i o n [62 1 ] of the r e s u l t i n g per iodic so lut ion generated the fo l lowing expressions for and r^Ct): A l l the numerical resul t s presented here were obtained i n double-prec is ion on an IBM 370 Model 168 computer, using a Runge-Kutta integrat ion routine with var iable step s ize and error c o n t r o l . • • 0.8 - • • O.t* -• 1 -0.2 I - o.t*. 1 0.4-• • -0-4 - • • • • - 0.8 - • Figure'6-1 • Periodic earth o r b i t about £ 9 = 0.012150, n 9 = 0.0 183 6.13 - 8 . 7 l t o 3 6 2 - 3 1 * I O " "^ cos . + 'I . 12-04- Z 2 0 Z 0 x 10 co$ 3 t (fc) = q. io$\ XI42,7 .* i o " q - 8 _ A . •V 1 . ^ 3 ^ 5 bb<\o"> * i o n^Zt -X.H10S 2 -42 -7^ * < o " \ 5 C ^ 3 t -8 + 6 . & o 3 7 S o i f c l * i o a*. 4 t - I.H6£> ? c j 5 ' 4 3 y; i o S i * . £fc ,.. . ( 6 . 3 . 1 ) -9 where terms less than 10 i n magnitude have been neglected. We now assume that Equations 6.28 and 6.29 represent an exact per iodic so lu t ion of the three-body equations of motion. The dif ferences 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 so lu t ion defined by Equations 6.26 and 6.27 are shown i n Figure 6-2 f o r the in terva l 0 ^ t < 2TT. The dominant errors are associated with terms involv ing cos 2t and s in 2t ; by comparing c o e f f i c i e n t s i n Equations 6.26 to 6.29 the percentage error of these terms i s found t o be 3.5256% and 4.4130% respec t ive ly . Both o r b i t s are plot ted i n Figure 0 < 5 -I IO 0 - 3 -J 181+ 6.14 -I t 2.TT S/>^  x io S 0 - S - , o-2S A O-XS -i .TT ZTT - O S -I Figure 6-2 Pos i t ion errors for the earth o r b i t over one period A r j x i o 4 ^ — • \ • \ . . \ > N \ / / . / . 0 . 8 - •\ \ . • / \ /'• • \ / \ /'• 0 . 4 - • \ / • 1 0 ' 0 . 8 - 0 . 4 0 0 . 4 0 . 8 Figure 6-3 Numerical and analyt ica l solutions about ^ = 0.012150, - 0.0 186 6.16 6-3 for the f i r s t quarter per iod, 0 ^ t ^ TT /2 , to provide some visual 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, discrete 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 in terva l At = 2TT/100. 6.2.4 Periodic orb i t s for 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 stars forming a close binary o r b i t a r e l a t i v e l y d is tant star of comparable mass. Configurations of t h i s type may be described by Equations 3.31 and 3.32 provided the condit ion l / 2 r « 1 i s s a t i s f i e d , and i n t h i s sect ion we invest igate the p a r t i c u l a r case defined ....by y^ = 10, ^ = 0-5 and r-j = 50. Thi? C h o i c e O f n a r a m p t p r s H p > $ p r \ / p s ^ n m p rnirmpnt.. F o l i a t i o n s 3 >/T. 4.101 to 4.104 w i l l e x h i b i t rapid convergence i f the quanti ty e t 5 6 i s smal 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 ) - • When y^ = ( l - ^ ) = 0-5, i t fol lows that e * 0.5/y^ 7 5" , and therefore e3y-|'/2~ - 0.125/y^. I f 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 per iodic o r b i t for p^ from Equations 6.1 and 6.2 as: S^(fcO - - 6 . 6 6 9 0 * io~fc - 4.. I S S o m o ? * lo~S c&t c f . page 4.32. 187 6.17 - 7 -/>f f t . ) .. - 8 - 4 " 5 o o 4 4 6 i 5 v. i o ' S sin. b I , 2 ? S "A I c " ^ S i r i 3 t + t .«43 ? 5 x lo~'° 4-fc . ( 6 . 3 1 ) The error incurred by neglecting terms O ( e ^ ) i s l i k e l y to be dominated by the quantity e u-j , 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 i s shown i n Figure 6-4, where A£ and An denote the respective displacements from % - 0.5 and r\ = 0; the d iscre te data points define the loca t ion of p 2 at successive time instants 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 quanti ty 11x^^(0) - x^(0) j| below 1 0 " ^ , and the fo l lowing per iodic so lu t ion was obtained: - 4 . i < ? 2 - 5 < » I 4?5 •,* to's cost 188 0 . 8 H 0 . 4 H 6.18 • 0 . 8 •0 .4 - i A ^ x l O 4 0 . 4 0 . 4 H - 0 . 8 Figure 6-4 Periodic o r b i t for .u, = 10, u 9 = 0 .5 , r , = 50 about C 2 = 0 .5 , n 2 = 0.0 189 6.19 - S . $ " 3 4 ? 4o$zo * | 0 co5 4 b ( 6 . 3 2 ) = g . 4 4 « ? - 6$||5 * io~.5 Si«.fc + S.6g<J3 IS3S5 * | o " ? Swy.Zfc . + . 1 . 3 3 1 6 bb$$o % io""' ' s ^ 4 b , ( 6 . 3 3 ) where terms less than 10~ I U 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 scale of Figure 6-4) are shown i n Figure 6-5. The dominant c o e f f i c i e n t errors 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 r espec t ive ly . 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 susceptible to e r r o r . 190 6.2C 0 - 4 H o -4 H - 0 - 8 H IT -I t 0 - 2 0 - 1 H o - i H 0 - 2 TT i t 2ir Figure 6-5 Pos i t ion errors for 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 , in 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 solut ions for the motion of p 2 and p 3 which s a t i s f y Equations 4.101 to 4.104. With, the solar system constants given on page 6.10 we have, from Section A4 .1 : = i . £ 6 7 4 at l _ 4 - - l . 2 t ? 4 ?o6£4- at U 5 % ^ =. - 4 . S 7 2 4 - 96€6o K. /o~' * . - 8 . 6662 54-04-0 * co~'- at L c b Note that v 2 and change in sign but not magnitude at Lg. The corresponding o r b i t near of the earth-moon system can now be determined from Equations 5.159 and 5.160 as: 1 S • 6&OZ. £ 4 0 4 ° * 10 192 g C t ) r - 4.8731 474^1 * 10 - i 1 Ct) = g. 65"2| 7 8 0 3 3 * io -1> 6.22 "5.?<iZl 4 * IO sc«. t 4 i . o68D ^ o z l * io" 6 <x>s t + S". 6?o8 84-639. v. i o " 3 sc^Zb * -«- 4- £547 4ST63S * i o " 3 co5 2b U - » 0 2 . 1361 *?4o58 x. io sc*-b fe I . 8 5 6 0 7 4 1 1 3 v. l o ~ Cost - 2 - 2 2 8 1 I«'j44 x i o " 3 $i*,Zt + 5". 6 8 3 0 7 o S 4 S x l o " - . . c o s 2 t : An equivalent o r b i t near Lg fol lows d i r e c t l y from t h i s so lu t ion when terms indicated by an as ter i sk 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 for a 2ir-periodic o r b i t using the algorithm out-l i n e d i n Section 6 .2 .2 , with Equations 6.34 and 6.35 def in ing 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 quanti ty _ i p l l x k + - | ( ° ) _ x|<(0)ll below 10" , and a f t e r a numerical Fourier i n t e r -polat ion the fo l lowing resu l t s were obtained. • 193 4~. 8 7 3 6 6 3 i ? 6 K ID' 6.23 - 9 . 9 3 3 4 - . 6<i74o * to s c ~ t . + 3 . 3 5 V S A-BbZ^ x IO -4 cos- t - 3 _ 3 + S- 63S2. 3 o x o 5 * i o s ^ Z f c 4 4.Z493 7 3 4 ( 2 - * i o coS 2-t 4 6 . o 6 4 o 2 o 6 4 7 x i o s;«.3t - 2.. 7o4-3 "37909 K I D co5"3t - 6 -<> + z.fc465 o i i o ? x i o s;*_-4--t — 1.48 23 O 6 3 o l * | o cos 4 b ( 6 . 3 6 ) 8. 87532. *. io" - 2 . 4 - 3 6 6 9 I Z 3 3 J< i o Su^. t 4 7 . 4 - 9 X 3 6 4 4 2 - 4 * 'O c o s t - > . . / . / . v > j J 6 6 T | y . i O S i i * . £ + i . « . B * 0 ' Y t > 1C l O C o 5 4 3 . 6 4 6 0 6 7 Z 7 4 v i o s i O b . 4 4 . 4-436 8 0 6 7 4 * »o ct>s3t fc - 6 - 1 . 3 6 4 - 0 . 3 S I S T - x i o s c ^ 4 t ~ 2 . 6666 S S 3 0 4 x i o c©s4t C 6 - 3 ? ; 4 . 2 7 3 6 6 3 | 7 6 n i o " - s S . 8 3 3 4 6 S I I 4 a "> ' s ^ t + 3. 3 5 < S 4 - 8 7 S 8 ^ l o ^ c e j . t - 3 5. b?*Z -bozoS t. \o ' n^.7Jt + 4 . 2 - 4 9 3 734.11 x io~ 3 co S 2.k - fc> 6 . . 0 6 4 O 2.142-i- * i o $ j ^ 3 t • - Z . 704.3 8 8 6 4 - 2 - * \o~ cbs 31 x. 6 4 6 4 <??p io K i o " 6 s;«-4-fc - . 1 . 4 8 X 3 Z5"9?-9 x l o " 6 c o s 4 f c ( 6 . 3 8 ) 19k 6.24 — i 1 * - 7 - 4 3 6£> 1 M 3 I * I O " * 5 c ^b - 7 . A H 2 -3 6 7 ? 7 2 ^ l o " S c o s h - 2 . ZZ-S*? 3 S 6 7 I ?t i o " 3 s ^ i t - 5. bb\ 8 + 3. 64-6C? £>6oZ£ •< to - 6 - 4 . 4 . 4 3 6 ^ 0 Z Z « -< < o " 6 c o S 3 f c - I . - 3 6 4 0 ? 7 1 7 I * 10' s i * . 4 b + 2 . 6 ^ 6 6 M 7 § Z . * l o " 6 c o s 4 f c O3 < 0 Terms less than 10" have been omitted from these expressions. The symmetry evident i n Equations 6.34 and 6.35 for o r b i t s near L . and L c 4 0 i s maintained here, although the agreement between c o e f f i c i e n t s i s not always complete. These s l i g h t i r r e g u l a r i t i e s are , however, more l i k e l y to r e s u l t from the ef fec ts of quantizat ion noise and accumulated error i n the integrat ion process than from any non-uniform system behaviour. The a n a l y t i c a l and numerical ly-derived 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 equi la tera l t r i a n g l e point ( i . e . ? - £^ = A£, n - = An) and discrete data points define the loca t ion of 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 solut ions over one period are plot ted 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 per iodic i n t = 2TT. Note, at least i n the present case, that the per iodic o r b i t defined by Equations 5.159 and 5.160 i s dominated by perturbations per iodic i n t = TT and d i r e c t l y associated with p-,. Terms per iodic i n 195 6.25 A n a l y t i c a l so lut ion Am x 10 <- t = o,TT , xtr z Figure 6-6 Periodic o r b i t s near L, i n the four-body problem 196 6.26 0 . 8 -t = 3 J . - 0 . 6 - J | t = 0,Tr,2TT A n a l y t i c a l so lut ion Figure 6-7 Periodic orb i t s near U i n the four-body problem 197 x t o " 0-4-* * 1..-o-z 0-4 J >7 O-l J 0 - 4 J TT ATT TT .aTT Figure 6-8 Pos i t ion errors for the o r b i t near O-br 5£ x I o " 1 198 6.28 o.i . H 0-4 J 0-4 - i 5*7 x io* 0-3. H • o - Z J rr ZTT T f — i ZTT Figure 6-9 Pos i t ion errors for 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 inf luence of p-j on the motion of p 2 and pg, and are r e l a t i v e l y small i n magnitude. Perturbations associated with p-j and per iodic in t = 2TT w i l l , however, be encountered when higher-order 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 s o l u t i o n s . I f , ins tead , we express the p o s i t i o n errors <5£ and 6n i n terms of. the maximum displacements a x and [An|m a x» then: |6C| * 0.048913 | A 5 | m a x i <5nI ^ 0.043673 An 1 1 max 6.4 Discussion Although the numerical so lut ions 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 cer ta in that the numerical and a n a l y t i c a l o r b i t s are equivalent . 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 fo l lowing way. When the i n i t i a l condit ions 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. S i m i l a r l y o r b i t s i n the four-body Section 3 .6 .2 , q .v . 200 6.30 problem may correspond to case 10 instead of case 9. No adequate reso lut ion of t h i s d i f f i c u l t y i s possible without a de ta i l ed analys is of motion involv ing more general perturbat ions. Comparatively few e x p l i c i t solut ions of the three and four-body problems have appeared in a form compatible with the resu l t s of t h i s chapter, which makes any fur ther comparison d i f f i c u l t . H i l l ' s s o l u t i o n for lunar motion i n the r e s t r i c t e d problem i s poss ibly the c loses t equivalent to the o r b i t defined by Equations 6.26 and 6.27, but t h i s case has already been discussed i n Section 4 .4 . An analys is of per iodic motion i n the general three-body problem by Moulton [63 ] appl ies to o r b i t s described as case 3 i n Table 3-1; more recent ly Arenstorf [64] has derived corresponding resul t s for case 4 o r b i t s . Unfortunately neither analysis 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 equal ly problematic, with only one o r b i t showing s i m i l a r behaviour to the so lut ion described by Equations 6.34 and 6.35. Kolenkiewicz and Carpenter [ 1 7 ] invest igated motion near of the earth-moon system for 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 larger i n s i z e , the r e s u l t i n g o r b i t exhib i t s some of the features shown i n Figure 6-6. Equations 5.159 and 5.160 indicate that the motion of p 2 and pg exerts comparatively l i t t l e inf luence on the four-body o r b i t (as discussed on page 6.29 )which may account for the success of the very r e s t r i c t e d model i n t h i s instance. Three re lated solut ions near should also be mentioned, although s t ruc tura l di f ferences i n the methods of analysis 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 di f ference of 180°; the d i sp lace -ment from i s comparable to the earth-moon dis tance , 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. S i m i l a r resu 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 per iodic one-month o r b i t s for 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 . A stable periodic o r b i t was found, smaller i n s ize than the numerical so lut ion described i n [30 ] and which corresponds to case 10 of Table 3-1. 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 : the general three-body problem and a r e s t r i c t e d four-body problem. Although a number of constra ints 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 consistent and ra t iona l approach to the analys is of four body systems has not previously been developed, and an attempt i s made here to remedy t h i s d e f i c i e n c y . The basic s tructure 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 fec t of more general perturbat ions . In the p a r t i c u l a r three-body problem under inves t iga t ion two masses, forming a close binary system, o r b i t a comparatively d is tant mass. A new a n a l y t i c a l so lut ion of t h i s problem i s found i n terms of a small parameter e, which i s re lated to the distance separating the binary system and the remaining mass. 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 ) and accurate w i t h i n a constant error O ( e ^ ) . Although many terms are l o s t i n the process, i t i s possible to reduce the ser ies so lu t ion 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 const i tutes a severe t e s t , but comparatively close agreement i s found between the two .orb i t s . The predicted so lut ion i s p e r i o d i c , a feature that allows further comparison with numerically-generated o r b i t s . Two examples are considered, the f i r s t of which re la tes to 203 7.2 the sun-earth-moon configurat ion of the so lar system. The second example appl ies to a problem of s t e l l a r motion where the three primary masses are i n the r a t i o 20 :1 :1 . In both cases the numerical and a n a l y t i c a l solut ions show close agreement, with an error below 5% for the sun-earth-moon configurat ion and less than 3% f o r the s t e l l a r 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 into the close binary system. Unique u n i f o r m l y - v a l i d solut ions are found for motion near both equi la tera 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 uniformly-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 with in a constant error 0(e ) , and the solut ions are uniformly v a l i d as E + 0 for time i n t e r v a l s 0 ( e ~ ^ ) . The predicted per iodic o r b i t s are compared with corresponding numerical so lut ions f o r motion near and Lg of the earth-moon system. Orb i ta l pos i t ion errors are found to be less than 5%, and i t appears l i k e l y that an extension of these resu l t s to the next order i n e would produce a substant ia l im-provement i n accuracy. Further work on t h i s topic could take a number of possible d i r e c t i o n s . The solut ions 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 . No d i f f i c u l t y i s forseen i n e i ther development. B.eyond t h i s p o i n t , however, the process of analysis i s l i k e l y to become considerably more involved . The s i tua t ions which r e s u l t from more general perturbations have been indicated 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 in describing system s t ructure . On a much more general level the ent i re approach may be used i n the analys is of non-integrable dynamic systems, p a r t i c u l a r l y when i t i s f eas ib le to decompose the problem into a number of subsidiary cases. 205 Rl REFERENCES 1. 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American  Mathematical Society Transactions, V o l . 7 (1906), p. 537. 64. Arenstorf , R . F . , New per iodic solut ions 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. H e n r i c i , P . , Elements of Numerical A n a l y s i s . Wi ley , New York, 1964. 66. F le tcher , R . , A new approach to var iable metric algorithms. The Computer Journa l , V o l . 13 (1970), p. 317. 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 212 r A i i TABLE OF CONTENTS Appendix Page I . EXPANSIONS IN TERMS OF e A l . l I I . THE REFERENCE LOCUS CONCEPT . A2.1 A2.1 Equations of Motion A2.1 A2.2 Numerical Explorations A2.4 A2.2.1 Intersections with the £ axis at t = 0 A2.5 A2.2.2 Posi t ions of equi l ibr ium when tfO A2.14 A2.3 Limitat ions A2.16 I I I . SOLUTION OF THE THREE-BODY PROBLEM A3.1 A3.1 The Homogeneous Solut ion A3.1 A3.2 The Non-homogeneous Solut ion A3.2 A3.3 Expansions of s in(t+0) and cos( t+6) A3.5 A3.4 Continuation of the Uniformly V a l i d Solut ion A3.6 IV. VARIATIONAL ORBITS IN THE RESTRICTED PROBLEM . . . A4.1 A4.1 The Homogeneous Solut ion . A4.1 A4.2 Non-homogeneous Solutions A4.7 A4.2.1 Non-resonant case . . . . . . . . . A4.8 A4.2.2 Resonances involv ing short-period terms A4.10 A4.2.3 Resonances i n v o l v i n g long-period terms A4.12 213 A i i i Appendix Page V. SOLUTION OF THE FOUR-BODY PROBLEM A5.1 A5.1 Expansion of the Nonlinear Terms A5.1 A5.2 Asymptotic Expansion of the Derivat ive Terms A5.9 A5.3 Continuation of the Uniformly V a l i d Solutions A5.16 V I . CHEBYSHEV POLYNOMIALS A6.1 A6.1 Introduction A6.1 A6.2 Function Approximation Using Shi f ted Chebyshev Polynomials A6.2 A6.3 Coef f i c ients of T k * (y) A6.4 A6.4 Approximate Expansions A6.4 A6.5 Expansion of the Function |r d i . | A6.9 A i v LIST OF TABLES Table Page A2-1 Astronomical Constants A2.9 A2-2 Parameter Values A2.10 A6-1 Chebyshev and Binomial Coef f i c i en ts for the expansion of f (x ) = [ l + x ] " 3 / 2 A6.13 215 A v LIST OF FIGURES Figure Page A2-1 Primary conf igurat ion at t = 0 . . A2.5 ,A2-2 Locus near L-j A2.16 A2-2-1 Polar representation . . A2.16 A2-2-2 £ component • . A2.17 A2-2-3 n component A2.17 A2-3 Locus near L 2 A2.18 A2-3-1 Polar representation A2.18 A2-3-2 £ component A2.19 A2-3-2 n component . . . . A2.19 A2-4 Locus near L 3 A2.20 A2-4-1 Polar representation . . . A2.20 A2-4-2 £ component A2.21 A2-4-3 n component A2.21 A2-5 Locus near L 4 A2.22 A2-5-1 Polar representation . A2.22 A2-5-2 £ component . . A2.23 A2-5-3 n component A2.23 A2-6 Locus'near.Lg • A2.24 A2-6-1 Polar representation A2.24 A2-6-2 5 component A2.25 A2-6-3 n component . . . . . . . A2.25 A6-1 f (x ) = [ l + x ] " 3 / 2 A6.10 A6-2 Region of v a l i d i t y for the expansion of f (x ) =• [l+x]-3/2 near L 4 and L g A 6 . l l 216 A l . l APPENDIX I EXPANSIONS IN TERMS OF e Consider the two expressions £ tot + j*, L%<-%) and C A M ) where £,n define the pos i t ion of p^. From Equations 2.43 and 2.44 the pos i t ion of p^ i n the £,n>£ system of reference i s given by t = - r cos t where t i s the independent var iable defined by Equation 3.30. We can therefore express the quantity as l >c , l r i b - / ^ s ^ b ) + 5 +1 Cfil.3) since 217 A1.2 2 2 2 A binomial expansion of Equation A1.3 w i l l be v a l i d when E, +n « r-j , and consequently the two quant i t ies defined by Equations A l . l and A1.2 can be developed in terms of the small parameter - L . The f i r s t few terms of t h i s expansion are: J*l *oSt + = £, cost - ^ t o s t l " J_ - 3. ( j c © s t - ^ $ l n . t ) which demonstrates the cance l la t ion of terms involv ing Ax (see pages 3.12 and 3.15). With e = — (from Equation 3.28), a f t e r some tedious , but s t ra ight forward, algebraic manipulation we obtain the fo l lowing expansions for Equations A l . l and A1.2 . C A i . r ) In, IV 1 X x • 1 1 8 8 5 (A(.£>) 218 Al .3 6 /». { 1 + £ L it> 4-& o Z f c 15 ^ 4 - t - - - / s«vb - i o £ «~-3b -3Z -1>S si~Sfc ? C o t <^3b - ^ 5 c^Sfc 4^- 4 4 5 tfc 32-+ 3|5 yi^5b ? 3?- J 64- iz8 15-8 6 V l r u l 3 - - 6 V« £ i J - <•{ [" I - 3<*,2.b][ ^ to »1* <L zb - tos <.4t It • -35 (^,3fc -5-2-219 •315 to^t 1-2-«C3fc - 'JJhS <C5fc 6 4 A1.4 G o b + IfS C j 3 3 t r — 3*5 Qrp 5"b 64- <3-S «*« 1 1- oCe'O When £>n define the pos i t ion of p 2 and Equations A1.6 and A1.7 take the form L J L * 8 i o / s . 3 6 ^ / ' - / O It 4- <6 7£- c o s b + Ijf5 cos 3b + 3T? cos 5k ( A I . * ) 220 A1.5 it — • fc'^O-fO 5 2 a 5 j 9 c o s t +• if c e ^ t 1 - V $ 3 si^fc + ]S - -* L-St«.3t oCO J 13 + 6 / i ' 3 ^ z t ) - 3v< siM.2-tr Oct) X b S i t 3_ si«.b + VS_ «;^3fc 8 « 2-^ s 0 - 3 « o 2 f c ) - 3 u . s S ^ a Z b ^ o f t ) iS_ s;«.fc •+- w£ s^'St +• 3iS Si«v £ fc 64- x-xe (A l . l ) it * e Hi X. cxTfe) t ' 3 11,6-fO £ 5."-5 [ 3. «.~b + is $ ^ 3 b 1 L * • * . - v / s 3_ Q r o b - _1S <te3t d ©CO 221 Al 15 h 6 J i 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 A 2.1 APPENDIX II THE REFERENCE LOCUS CONCEPT1" A2.1 Equations of Motion The analys is 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 resu l t s of t h i s approach are summarized here to indicate the l i m i t a t i o n s which were encountered. These two paths of inquiry s t a r t to diverge at Equations 2.47 and 2.48, which describe the motion of p 4 near p 2 and p 3 i n the ri plane. We have, for 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 condit ions for r e l a t i v e equi l ibr ium at a s p e c i f i c ins tant of t ime. When the same conditions are applied 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 posi t ions 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 equi l ibr ium w i l l be c a l l e d a reference locus . + T h i s material was presented at the 13th International Congress of Theoretical and Applied Mechanics, Moscow, August 21-26, 1972, and at the 23rd International Astronaut ical 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 acce lerat ion of p^ are both zero, we obtain L E I * This device was mentioned, although not inves t iga ted , 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 i n 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 solut ions can be found i s described i n the fo l lowing s e c t i o n , but for 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 ( A » - 0 where a i s a normalizing constant, so that x and y define the motion of p 4 with respect to the .reference locus . Equations A2.1 and A2.2 can then be wr i t ten 2214. A2.3 - <v a. +• \ 7 4 — 1 IT- | 3 - 5 , Z— i i _ i 3 - a -<. = ( lit (A x.S) I f the term ' i s expressed as L I 3 i n - 1 J , r . , - J 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 involve terms i k of the form x J y ( j , k = 0 , 1 , 2 , ••••). When the o r b i t a l parameter a i s s u f f i c i e n t l y small these equations may be wr i t ten 225 A2.4 ( A * . 1 3 ) where a , b and c contain t e x p l i c i t l y . I f X Q and denote so lut ions of Equations A2.12 and A2.13 when the terms 0(a) are neglected, the approximate solut ions of Equations A2.1 and A2.2 are given by 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 A2.4. A2.2 Numerical Explorations At a p a r t i c u l a r instant of time the pos i t ions of r e l a t i v e equi l ibr ium are defined by the condit ions S = % u + °-*o (A2.I4) 1 * "lc (R2 . . IS ) 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 possible to predic t i n advance the to ta l number of reference l o c i . D e f i n i t e resu l t s can, however, be obtained in one s p e c i f i c instance, and the approach i s described below. A2.2.1 Intersections with the E, ax is 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 A2-1 . Figure A2-1 Primary conf igurat ion at t = 0. Note that t h i s i n i t i a l conf igurat ion i s chosen so that both p-j and p 3 are located on the negative E, a x i s . One so lut ion of Equation A2.17 i s thervf (0) = 0, and we have - X 227 A2.6 This equation can be wri t ten ( s , - O x f ^ - O " <* , -0* from which a seventh-order polynomial i n E,^ can be der ived. The c o e f f i c -ients of t h i s polynomial depend on the signum functions sgn(£.j-£;^), and i n f a c t there are four d i s t i n c t cases corresponding to the condit ions When Equation A2.19 i s expanded we obtain — 228 A2.7 + 5 t -1, + sC-c 6C7-) + ^ f ^ s r ( 5 , - O +• fx-S^^C^- O = o (fli .^o) where &(4> = - 2. 229 A2.8 We must apply numerical methods to obtain so lut ions 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 2 and p 3 , suppose we take the model described i n Chapter 2 (Section 2.6 , on page 2.30 ) , with the addi t ional r e s t r i c t i o n that p 2 and p^ move 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. The l i m i t a t i o n s of t h i s approach have been discussed i n the introduct ion to Chapter 3, but any e x p l i c i t so lu t ion for 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, consistent with these r e s t r i c t i o n s , can be derived from the fo l lowing expressions: X = r Z3 n * = X 3 L S r, = At X C = r, - ( jry hx R 3 H 230 A2.9 Parameter Symbol Value Astronomical Unit A 0.4990 04785 x TO3 l i g h t sec. V e l o c i t y of Light c 0.29979.25 x TO9 m. s e c . " 1 Ephemeris Seconds Per Tropical Year (1900) s 0.3155 6925 9747 x 10 8 sec. Earth-Moon Mass Ratio . m 2 /m 3 0.813024 x 10 2 M o a n F s r t h - M n n n l l i c t a i i r o PI Geocentric Gravi ta t ional Constant * r Z6 •: IT Km2 q n 3«4-inn y i o m 3.1415 9265 3589 79324 0.398603 x 1 0 1 5 m 3 sec? 2 Table A2-1 Astronomical Constants 231 Parameter Value C 0.3891 7127 26 X 10 3 r l r 2 0.3891 0.1215 7245 0313 58 96 X X 10 3 ID" 1 r 3 0.9878 4968 60 n 1.0 0.7470 2544 92 X l O " 1 y l 0.3289 2414 85 X 106 y 2 0.9878 4968 60 y 3 0.1215 0313 96 X l O "1 Table A2-2 Normalized Constants 232 A2. where the quant i t ies A, c, s and Km^ are defined in Table A2-1 . The numerical values for the astronomical uni 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 for 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 for the parameters of Equation A2.20 have been obtained, we must se lect a numerical method f o r so lv ing n ' t h order polynomials with real c o e f f i c i e n t s . Rutishauser 1 s quot ient-di f ference algorithm [ 6 5 ] 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 to a l l roots are derived simultaneously. There are , i n f a c t , four d i f f e r e n t polynomials to so lve , corresponding to the cases defined on page A2.6. Roots of Equation A2.20 for these four separate cases were found using a version of the quot ient-di f ference a lgor i thm, and the so lut ion accuracy was tested by evaluating the polynomial with these s p e c i f i e d numerical values . The resu l t s are given below. Case 1 £ L > 0.12150 31396 x 10' •1 REAL O. 39 O 2. S"8 3 4 -*. i o 3 0.39 02. S 8 3 4 x i o " O . 1 3 6 3 6 6 3 2 0 . 1 8 6 3 6 6 3 2 . O . 490 0 2 8 ? € O .4-9 o o 287-8 O . IOO l 3 9 I 3 x Io' COMPLEX 0.2.905 I 8 4 1 x i o 1 - 0.7-<\OS 18 4 2 K IOZ O. 7 - 9 4 9 l 9 9 o * i o " 1 O. 7 - 849 19 9 0 * i o " ' 0 . 8 S 5 7 I l o I O . 8557 If o I O.o 233 A2.12 Case 2: -0.98784 96860 < £. < 0.12150 31396 x 10" 1 REAL COMPLEX - o . 3 1 0 2 . S"834- O . 2 -1 0 s l S 4 2 ( o 1 -- o. 3°02- S"8 3 4 X i o 3 - 0 • Z 1 o 5 ~ o. 1 D 6 2 . 5 ? o 3 101 O - I 4 ? 3 31 3 2 - o. | 0 6 2 5 7 o 3 < 10 ' - 0 . I 4 S 3 3 l 3 8 -o. 1 0 6 2 . 5 " ? o 3 x . o 1 0 . I ft S 3 3 1 3 8 - o. 1 0 6 2 S?oS .0' — 0 . I 4 - S 3 3 1 3 g - 0 . 8 3 6 0 8 3 7 2 , 0 . O Case 3: -0.38971 24558 x 10° < REAL - o . 3<jo2 < t o " 5 - o. 3 l O i 5834- * vo3 - o . l o zz T - i 1 I - o . <\07-Z. O . S03S 4-581 O. 5o3S A -5SI - O . I l b 3 <?4 -8o * 10 1 Case 4: ? L < -6.38971 24558 x REAL - O . 35€« o 2 1 I * 10 3 - o. 4 l ? l Zo3fc x io 3 - o. 4-44-4- 73S"S x 101 O. 3627 14-74 x io~' o . 36. 8? I 4 74 x io"' - O . <?8 6<? o o $ l - o. 1861 o o $ i KL < -0.98784 96860 COMPLEX 0. X°ioS K 107" - o . Z l o S " 184-2. x IO*" O . I 3>06 7 < ? 7 8 - o . 1306 7 1 7 8 o . ? 6 4-8 - o . 8 6 04- 7 6 4- 8 O - O 10 3 COMPLEX 0 . 0 o . o 0 . 0 O . 4 7 2 k 3 © 4 I - o . 4-72-6 3 o 4 - l O. 5 Z 8 3 o 1 6 2 K 10' - O . S Z 8 3 O l 6 Z xi 1 o 23k A2.13 -15 In a l l cases the residual error was below 10 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 ther zero or a wel l -def ined 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 wi th in 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 that we can state the f o l l o w -ing r e s u l t : -When t = 0, for the p a r t i c u l a r numerical model chosen here prec i se ly four reference l o c i in tersec t the £ a x i s . The points of in te rsec t ion are € = 0.1001 3913 x 101 K = -0.8360 8372 K = -0.1153 9480 x 10 1 € = -0.4173 2036 x 10 3 . I t might be ant ic ipated that three of these intersec t ions w i l l be located near c o l l i n e a r Lagrange points of the corresponding r e s t r i c t e d problem. From Szebehely [ l 1 8 ] , when = 0.01215 the c o l l i n e a r points are located (approximately) at £ = -1.15568 (1^) K = -0.83692 (L 2 ) C = 1.00506 (L 3 ) , so that the conjecture i s j u s t i f i e d . This r e s u l t can, however, only be applied when Equations A2.16 and A2 . 17 reduce to the form of Equation A2.18, which i s true only when (n - ft)t = 2kir, where k = 0, ± 1 , ±2 , •••• . 235 A2.14 A2.2.2 Posi t ions of equi l ibr ium when t f 0 The functions f^( t ) and f ^ U ) i n Equations A2.16 and A2.17 define two nonlinear algebraic expressions, but the formulation i s not p a r t i c u l a r l y h e l p f u l . I f , however, we take then F(£ , n , t ) > 0 and a so lu t ion for £ and n which s a t i s f i e s the condi t ion at a s p e c i f i e d instant of time w i l l a lso s a t i s f y Equations A2.16 and A2.17 at t h i s time ins tant . Fletcher [ 6 6 ] has developed an e f f i c i e n t algorithm to determine the loca l minimum of a nonlinear algebraic funct ion of n variables i n the v i c i n i t y of an i n i t i a l est imate, which can be applied to solve Equation A2.22 for 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. At t = 0 select an i n i t i a l estimate of the point on a p a r t i c u l a r locus . From th i s estimate, determine the exact l o c a t i o n of the point using F le tcher ' s method to solve Equation A2.22. Increment t by A t . With the preceding point as an i n i t i a l est imate, i t e r a t e the process by returning to step 2. 1. 2. 3. 4. 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 Equation A2.22 for every value of t . There i s some evidence to suggest that F(£ , n , t ) = 0 cannot be s a t i s f i e d for a l l values of t along two l o c i , but a discussion 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 A2.3 . Six l o c i were found for the sun-earth-moon system by t h i s method, using the approximate primary model of the preceding sub-sect ion. For each i n d i v i d u a l point the funct ion 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 pos i t ion correct ion was less than 10 in magnitude, -20 which in a l l cases reduced F below 10 . Four of these l o c i emanate from the points of in tersec t ion given on page A2.13,and the remaining two l o c i correspond to equi la te ra l t r i a n g l e points of the r e s t r i c t e d problem. The locus containing the point £ = -0.4173 2036 x 10° , n = 0 forms a c i r c l e in the £ , n. plane so that the point of r e l a t i v e equi l ibr ium 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 for the f i v e l o c i near the earth-moon system are shown i n Figures A2-2 to A2.6 . Note that the two l o c i near L-| and 1_2 are s i m i l a r i n character , and so also are those associated with and Lg. The locus near L ^ , shown i n Figure A2-5 , 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 into con-s idera t ion the e f fec t of so lar 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 l 1 -\-\SS A2.16 1. - 1 - 1 5 4 h -0-2 0-4-Figure A 2 - 2 Locus near L^ A 2 - 2 - 1 Polar representation I 238 A2.17 - 1 - 1 5 4 - - r •1 -tSS 1-156 OS 1 t Figure A2-2-2 E, component 0-? -1 o -is -4 o-S o-s -J Figure A2-2-3 n component 1 t. 1 -o Figure A2-2 Locus near L^ 239 - 0 . 8 3 7 - 0 . 8 3 6 A 2 . 1 8 r 0 . 2 h o.i V 1 0 - 0 . 1 *• - 0 . 2 Figure A2-3 Locus near L, A2-3-1 Polar representation 22+0 A2.19 5 L - 0 . 8 3 6 i - 0 . 8 3 7 H - 0 . 8 3 8 r — > • . t 0 0 . 2 5 0 . 5 0 . 7 5 1.0 Figure A2-3-2 £ component 0 . 2 0.1 1 0 ' - 0 . 1 - 0 . 2 J 0 . 2 5 0 . 5 0 . 7 5 - t 1.0 Figure A2-3-3 n component Figure A2-3 Locus near L 1.0 214-1 A2.20 0 . 7 5 H 0 . 5 0 . 2 5 - 0 . 2 5 - 0 . 5 - 0 . 7 5 - i 1 0 . 5 0 . 7 5 1.0 1 . 2 5 - 1 . 0 Figure A2-4 Locus near L^ A2-4-1 Polar representation 1.0 0.8 0.6 0.4 2i+2 A2.21 0.25 0.5 -T- ' t 0.75 1.0 Figure A2-4-2 £ component 1.0 1 0.5 -0.5 H 0.25 0.5" -1.0 J 0.75 ^ t 1.0 Figure A2-4-<3 n component Figure A2-4 Locus near L 3 0 . 8 0 . 7 0 . 7 • 0 . 5 Figure A2-5 A2-5-1 - 0 . 3 - 0 . 1 Locus near L^ Polar representation 0.1 0.1 1 0.1 i 0.3 1 0.5 1 A2.23 0.7 1 1 1 1 1 t 0 0.25 0.5 0.75 1.0 Figure A2-5-2 E, component 1.0 n 0.9 0.8 0 .7 0.25 0.5 0.75 - \ t 1.0 Figure A2-5-3 n. component Figure A2-5 Locus near L^ 1 L •0-7 o-S -o-3 -o - l o-i -0-7 H -O - S l - O ro Figure A2-6 Locus near L^ A2-6-1 Polar representation ro -p. - 0'\ 0-3 i -OS - 0... 7 2i;6 A2.25 — i 1 r - 1 t .0-2.5 0 - 5 0 - 7 5 i-o Figure A2-6-2 £ component - 0 . 7 - 0 - 8 -o.<\ J - i -o H O'XS o-S i t • o Figure A2-6-3 n component Figure A2-6 Locus near L, 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 per iodic functions of time. A2.3 L imita t ions One p o s i t i v e r e s u l t of t h i s analys is i s the extension of Tapley and Lewallen's inves t iga t ion [ 9 ] to include a l l pos i t ions of r e l a t i v e equi l ibr ium for the sun-earth-moon system at a p a r t i c u l a r time ins tant . A number of features , however, render the reference locus approach unattract ive i n the analysis 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 tered the ent i re analysis must be repeated, ( i i ) When the o r b i t a l parameter a = 0, the functions £ ^ ( t ) and r i^ ( t ) are not solut ions of the equations of motion (because the quant i t ies E,* and i n Equations A2.7 and A2.8 are 0 ( a - 1 ) ) . Although a may be s m a l l , the l i m i t process a -> 0 cannot, therefore , be used to j u s t i f y neglecting terms 0(a) i n Equations A2.12 and A2.13. ( i i i ) 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 so lved, p a r t i c u l a r l y i f solut ions are to be accurate over large i n t e r v a l s of time, ( i v ) Not only the locus , 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 d i f f e r e n t i a t i o n 214-8 A2. i s i t s e l f s t ra ight forward, 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 . This behaviour i s evident in the two l o c i near and Lg (Figures A2-5 and A2-6) . An attempt to continue the l o c i through these abrupt t r a n s i t i o n s did not succeed because no so lut ion for 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 F le t cher ' s algorithm f a i l e d to f i n d a loca l minimum, but i t appears l i k e l y that no so lut ion of Equation A2.22 ex i s t s in these regions of the n plane for cer ta in 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 a r r - n m u l a t i ' r i n rt-F r l i f f i n i l f i abandon the analys is involv ing 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 der ivat ions (which are given below), the analysis of Chapter 4 up to Equation 4.83 requires l i t t l e fur ther a m p l i f i c a t i o n . We now continue the development beyond t h i s point to / -14\ obtain solut ions uniformly v a l i d for 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 wri t ten in the form o 3 o o I o o -X o o o o o X I o ( A 3 . f ) which i s equivalent to ^ = A where x i s the vector (x-|, x 2 , x 3 , x^) . I f x(s) denotes the Laplace transform of x ( t ) , then from Equation A3.2 we obtain - i 5E.Cs:> = ( s i - A ) 51 C o ) . CM. 3-") Now det(sI-A) = s (s+1), and a f t e r some straightforward manipulation: 250 A3.2 01-/0" 4 - 2±_ S--r\ z s o " 3- _ **•*•• 6 - 6 2s - 2L J _ . A- - 3 *Vl sl+r S S 6$ s - 2. o 4-S - 3 _ 0*3.4) Taking the inverse transform of Equations A3.3 and A3 .4 , we can wr i te x C t ) = <f>tl) x C o ) , where 4 - 3<<r>t 3 *.vfc Cos b 0 . - . 2.<<ofc X 4*J^b - 3 1 (A3.S) Equations 4.13 and 4.14 now fo l low d i r e c t l y from Equation A3.5 . A3.2 The Non-homogeneous Solution 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. I f B denotes the vector (0, E * , 0, F * ) T , Equations A3.6 and A3.7 can be wri t ten i n the form 5. = P5c •+ B and therefore s£s) = _ ( s J - A ) " V x ( ' o ) + (si -A)'1 BC*s"> where B(s) i s the Laplace transform of B ( t ) . When B(s) i s known, the non-homogeneous part of the so lut ion corresponding to the term (sI-A)~^ B(s) can be derived from Equation A3.4. periodic terms involv ing wt are considered separately. I f e ^ ( t ) and f^l^(t) denote the non-homogeneous solut ions of Equations A3.6 and A3 .7 , then for I t i s convenient to p a r t i t i o n Equations 4.41 and 4.42 so that I F * Cfc) +• b,b . t . b ^ t . + >j«ot. we obtain 4- ( Zb,-2b^-a 0) ^ *b 3-b s;».b (fl 3.8) 252 A3.4 - [ + 3 J . J l> - fe, t 3 _ L * J x + ( ftt + 2 - 0 b *~b + ( - * - 0 t *=,b Cft^O S i m i l a r l y , when the corresponding non-homogeneous solut ions are: [ b + :, - %u3d~Q cost X3L{ + ot 0 I su.«jb - %3-t L C o x - 0 , cJ b (A3, io") - 3 f 0 b to (B3 . i l ) ZkJC 0 + 41. onb ItOC. - ( 3 + w 1 5'' ifa>C0 + (.3> + «J cos <~>h 253 A3.3 Expansions of sin(t+8) and cos(t+8) From Equation 4.32 the expansion for 8 i s of the form A3.5 so that we can write ««. ( b + P) = suvb or>9 + c«fc + <i,fcJVs^P<> <*> CsQ, + e"1 e ^ . - - ) + ctr>Qt> ^ C e P , v t ^ P ^ .. . )J and, s i m i l a r l y , - COS - 5.K b ooCfeP, ^ ^ P , . * ••• > - s : ~ P 0 s i^C t P , ••• *) j b £ * ~ P 0 co C eD, * G"10i_+. • ) -t- Cr,t904^C 69, + t i e v + - •) ^ 0*3.13") 2 2 The functions sinCeS-j+e e 2 + * * * ) and cos(e8.|+e 8 2 «««) can be expanded as •51 1 - e^f* z! .(AVI4) CA3-15-) 25k We therefore obtain for sine and cos9: A3.6 8 X 9 0 - e,"1 ,^PC - I P x ^ 0 o + e ( x qo 00 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 Solut ion 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 • i t 1 (fl3.l«) 255 if- it = - X ' 5 + ^ . 5 OCe 1 it I it i\5 it?-+ z ^ V A3.7 (A3-19) s "f OCfc1") 2 ^ 5 7 - Z^,„ - 3 a l e = - Z - 2v s - - -itZt; OCe*) J O C t L x o - j - o . O C t ' " ) [ 1 it""- J oCc>) 1 if*- °t itz-c i r i o o ) Ofl) J (A3, io) - 2 d*V$ -+• 0)0.5 "(3.XV-) J <?Cfe*) + p . <'-/**) J OCt'°) o C t ) (fl*.ai> 256 A3.8 it - • i — — + 5" }\*? + bj*s~l +5 - 7 X iC i t J ^ J ^ ) 1 at1- ^ Joco -_ it* • - 2 +- — b <js . itir L i t * at ^ J o C O I J b * 5V Joco J OCi) it* it 257 L Z G - ^ O J o C e ' * \ ) z - — i^J-f O C i 6 ) 1 Xt-*" Xt ) o C 6^> 1 it* bb Ttir bV J o t f e 3 ) I bt • d^Socrf it Sou) lit*- it J A3.9 oC.) J it L 0 - » o J ° ^ " 0 z — . «3b^r IV J OCeO '.•I. L * t * a t j o c £ < 0 (_ i t ^ i b a tar of t 1 . Li jfc C A 3 2 5 ) ^ , 5 - 2 - ^ ( > dfc»- it - 3u 258 A3.10 _ I it joceO L .dt* ^ ^ i it" at ^ j ^ ) L ^ ^ JoCe^ i t J o^) L a t * a t J o c o ©CO £ y l 3 + at* i t - F - 2. i t 1 I J > ) t J o f t o i a t * a t ^a-i? a r LTb* • 5 r v i 0 a O L a t - .at J &a T) £. «fc J L <>tv ht oCO J ; ' l ^ J o ( 6 ' ) •' L a t - •• aba*} o o o t- d t l" . . ^ J o c o The quant i t ies and F. are defined i n Appendix I , Equations A l . 8 and A l . 9 , and f o r . E g and Fg i n Equations A3.18 and A3.19 we have E g = Fg = 0 . Now consider the remaining quant i t i es i n Equations A3.18 and A3.19. 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 obta in : - X 0>\ts — ibiv bx 259 bx A3.11 2 t - Z 5 . + ^>JLg obtr bv which can be rearranged in the form: S*- bv (b+£>3/>_ + 2t? a-cr 2 -.1 - 2v s = 2 s.«. t at Zcos t ^ f cose 2 - 5 ^ - 6 1 ^ 5 z b v/s •+. ^ a5• bfi_ Si^9 <r b"C -2-cost ^ (b COS 8 - ph9 SLVLB 1 At the present point in the analysis we have shown that = 0 ( e ) , || = 0 (e 3 ) , f j = 0(e 3) and | f = 0 (e 2 ) . With these s i m p l i f i c a t i o n s : 3x 9T QoP - p>z® ~bt bV 5 9 ^ *«. 6>. +6* i 3/>, -f>0^bK -/j.e.af* -/s 0p , 5p 3 < C«0 Po I ar ^ 2. b 260 A3.12 3& t c*>9D + e • +6* l }v ytf zx 2. 3 Equations A3.28 and A3.29 can now be evaluated, and hence - 2- 2 5 ^ , - 6 t ^ CA3.32) + 2 cob" [ I to [ t CA3 -33 ) + 2s;*.fc 5" <?/s" -/S.P,5fix 0o + ] 261 From Equations 4.25 and 4.26: - 5 ^ t 4- S'". 8 5 - - e l A3.13 (G3-34-) 2-^ •Zy~!r + Ze^fc Six-do i I 3 t £fk 1 (.AS- 3 5 ) I and we tnen obtain A F ] /*. F I t has already been shown that | | i i s constant (on page 4.20) and conse-quently both these quant i t ies are zero. 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 : (A3.36) — 3 v"5 bt - ^ v - C t + p ) (ft3.3?) and from these expressions: 262 3.14 2fu.$ - Pvs ib> it oCt) (A-3. 3S) ( A 3 . 3 1 ) The terms I at it? a * J and 1 2 a V s -4- }u.s ~) are both 0(e), L at-av At? J . A Vand therefore do not contribute to the non-homogeneous s o l u t i o n of Equations A3.18 and A3.19. For Ug(t) and v g ( t ) we have (from Equations 4.60 and 4.61) : «?V6 - Xrb it* i t if1- it iVo Zb 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: _ * = %Hy(\+H>VC\-^')\ £ c»2.b -£ \ + 6 ( i + / 0 \ « * , + zip, - 6b£^ i-v I a r J L ai? iv 3 263 A3.15 + Ceo t J I a i r + * 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 test for error in these funct ions . 26*4. A3.16 2 I f terms involv ing t are to be el iminated from the non-homogen-eous s o l u t i o n , from Equation A3.9 : o which j u s t i f i e s the assumption that i s constant (see Equation 4 .82) . We have already shown that s i n 0 Q = 0, and with th i s condit ion on 6Q terms containing t s i n t and tcost are e l iminated from u g and v g i f i v «»9o\ 0+fO'>> - p0W* L iV d~C (A3 -4 -0 From Equation 4.76: i t (A 3-4?) and therefore Equation A3.45 takes the form X v 16 ( A 3 - 4 S ) The only value of 9^  for which 3g does not v i o l a t e the uniform v a l i d i t y condit ion i s consequently 9, - o CP 3 . 4 0 i n which case air (A3 r s-o) 265 A3.17 Equation A3.46 can be rearranged as and i f we again use Equation 4.76: • since cosBg f 0. The quant i t ies BQ»8-J and 6Q are a l l constant, and therefore <^ 3 = constant . At present we have no way of def ining 3 N > 3 N , and , although these quant i t ies are known to be constant. I f , however, i t i s assumed that both 9 2 and 6 3 are constant, then from Equations 4.81 and A3.49 we obtain where s in6 Q = 0. The one remaining secular term in v g ( t ) must e l iminate the quanti ty - e 4 ( 3 t a 4 ) i f v 5 ( t ) i s to be uniformly v a l i d to 0 ( e 5 t ) . Consequently, from Equations A3.9 and A3.11: 266 A3.18 and, i f we subst i tute for the quanti ty tos 9 1 / ^ f ^ + /*, ^ P r i from Equation A3.46, t h i s can be wr i t ten as From the argument of page 4.15: I t i s , i n f a c t , legi t imate 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 i s constant-E x p l i c i t expressions for Ug(t) and v g ( t ) 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 solut ions for u-j 2 and v-j 2 are considered. Now we proceed with the so lut ion of Equations A3.20 and A3.21 for u 1 Q and v 1 Q . This port ion of the analys is 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 and 'AI . 9 : 267 A3.19 ] (.A3. SrO L g |6 ( 0 3 . 5 8 ) and from Equations A3.28 to A3.31 we have, with 0^  = s i n 0 Q = 0: COS I + 2-eosk C<JD 9, ->f A > ^ /^>x^ 2- 1 ar a a t; _ £ /j0as<, +/\a^2 11 av ' PT- ' j . (ft - 2 cost 2 2o General expressions for o*<is and d V * are given i n Equations A3.34 and A3.35. At present, however, | | = 0 ( e 4 ) , | f = 0 ( e 2 ) , 0(e 4 ) and = o(e 4 ) , so that ]>jf$ and }j*s are both 0 ( e 4 ) . Consequently orz a t * bxx i t fo l lows that : * o 0Ct») oU1) 268 From Equations 4.47 and 4.48: -t- o a 6 ) A3.20 (ft3.6l) (A3.62) since 0-j = 0 and a Q = 0 (see Equation 4.52) . The nonlinear terms i n Equations A3.20 and A3.21 can now be evaluated, and we obtain • a ( 3 / - - feu.*) - 9 /* cos 2.b / * o  ( A 3.63.) I (l- /*>) (see a lso pages A3.26 to A3.39 f o r a de ta i l ed d e r i v a t i o n ) . When sin0Q= 0 and 0.| = 0, the equations for sin0 and cose s i m p l i f y t o : «~ 9 O c V ) (A3.65) L 2. . Cos » 0 269 (see Equations A3.16 and A3.17 on page A3.6 ), so that s i n ( t + 9 ) and cos( t+9) can be w r i t t e n : A3.21 s.v. Ct + B) 5~t | I - £*fC ^  + cosh | ocv; CO ft, We then obta in , from Equations A3.36 and A3.37: (A3.iS) ^b* a t r ~* — (ie e^,90 east +- £ |^  3<*, - <<o b J C « 0 t - *~b 270 A3.22 3b* at L 4- 0 ( t f ) , and therefore cIVj - a>/s ^ a t 3 o i x - &o - <*>9, 033-92) From Equations 4.25 and 4.26: dldX z. - a j / 5 = -at a»t a - r atr a r (A3.7S) 7- S~^s + a_t< 5 a t a r a r % f / 3 ^ t + O - 3 / i s i ^ C t + a" ) - A- <u ' a -r <ar a_u. $ 2 ^ ; dx~ L a-u"1- 3x - 3 t a ~ixr a - c j a r * L a r 2to$Cb-*-e) 2 s ^ b + e ) (A3.?6) 271 A3.23 At the present stage of the a n a l y s i s , in addi t ion to the order r e l a t i o n -ship given on page A3.19, we have ||> = 0(e 4), = 0(e 4), 0(e4) 9x and ^lif = 0(e ) . None of the expressions defined in Equations A3.74 d r 2 -to A3.76 therefore makes any contr ibut ion to Equation A3.20 and A3.21. From Equation 4.69 u 7 ( t ) = Vy(t) = 0, and also (see Equations A3.40 and A3.41) , so that we can now accumulate the non-homogeneous terms in Equations A3.20 and A3.21. The corresponding funct ions , E* and F* , are given below. 6 b L ar L * r J d - ^ O J ' + o o b" C«>9/ atr ' a t ' L iX + . Q o Z b 1 f*,(>-fO 4- It, :* - 2. 9^-^ IT 1 ^  i TTT^ ( A3.?« ) - 35 YX C<- S£~4-b 272 A3.24 To el iminate secular terms involv ing t from the non-homogeneous so lut ion for v 1 0 : - ° (A3.rO which j u s t i f i e s the assumption in Equation A3.56 that i s constant. Terms containing t s i n t and tcost are el iminated from u ^ ( t ) and v ^ ( t ) i f : - f>o^->-\ =• ° (AS . s o ) L a r J f>„)K + C^—y + fc\ ^ ' ? - ° • ( A i . s O ar br I >r J We can again subst i tute for the quantity iP> - ( l + ^ ^ ^ J from Equation A 3 . 4 7 , and Equation A 3 . 8 0 then takes the fo l lowing 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 Qz = O (A3.S3) the function 8^(x) v io la tes the condit ion 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 . 8 4 ) 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 £f>3 are a l l constant: H.* = constant . (fB.sO I f we now assume that 8 3 and 6^ are constant;, then from Equation A3.85: (see Equations A3.170 and A3.211). The remaining secular term in 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 that a_ i s constant (see Equation b A3.167), t h i s secular term must el iminate the quantity -e 5 (3tag) i n 27k A3.26 the expansion of V c ( t ) . From Equations A3.9 and A3.11 we then have: By the usual argument (page 4.15) : O , (A3.S8) and, a f t e r subs t i tu t ing for BQ from Equation 4.81, Equation A3.87 takes the form 31 + 2ZS" S12-C1 + 0 Note that Equation 4.81 f o r 3 Q i s now known to be v a l i d (since = 0). and consequently the spec i f i ed value of 3 Q can leg i t imate ly be used i n the a n a l y s i s . Before proceeding with the so lut ion of Equations A3.22 and A3.23 f o r u ^ and v ^ , i t w i l l be helpful to evaluate the nonlinear functions 2 - C l - f O and e ' 2 _ C3u.v^ i n d e t a i l . This process (which i s rather tedious) i s only continued to 0 ( e 1 4 ) . 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 cance l la t ion of secular terms in v.-(t) o i s taken into considerat ion. In t h i s case the quanti ty - e ^ t o u ) i s omitted from the expansion for Vg( t ) , and corresponding secular terms in v^ + . j ( t) are ignored. The p r a c t i c a l consequence of th i s s i m p l i f i c a t i o n i s tha 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 tc . 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 cer ta in order, and to be precise we should w r i t e , for example: L « J 1 '2. 3 * J 276 A3.28 From Equation 4.69 u 7 ( t ) = v 7 ( t ) = 0, so that we have: Each term in these equations must now be evaluated. For u ^ ( t ) , from Equation 4.25: and we obtain Now, from Equations 4.52 and 4.75: and therefore The expansion for 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 +'' 8 T3 / *o ] + 0 . ( & * ^ • CA3.ID2) When G 2 = 0, Equations A3.67 and A3.68 for s i n ( t+9) and cos(t+6) reduce to i^Ct-h9y = cas90 si^b + 9^ CPS0 O c o s t + O f f c O ( / u . i o j ) cos cos &0 cost - ez 6?3 to$90 s ^ f c tXfc*-") , From Equation A3.104 we therefore have which may be wri t ten c^Cfc + e ) > _L ( i+• coZb") - e ^ s ^ z b -t- o f t 4 } ( A S . I O O 2 since cos 6 Q = 1 (see Equation 4.78). We are now in 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"1^ 4- et,1" + + / 3 ^ ^ ( l f o o Z t ) - 4- * t f i t t n 9 0 cosh J + O f t * ) . CflrioO From Equation 4.60 for u g(.t) we have: 2/*|Cl-^i - /3<to(b + e) ^ j" 2 Cot - Cco^ b - 1 j , (ft 3.107) and using Equations A3.98, A3.100 and A3.104 th i s can be wr i t ten 2 n.j l i j = 2 . ^ , 0 - / 0 ^ /3 0 «r>90 £ - 1 + 1 co 5 b - coj2fc + ± cos 3 b + & £ 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 .83, A3.100 and A3.104 we obta in : u.. 279 l t v r > 4 t 2-A3.31 (fl-3.|0<j) 1^  87 - 7 coot +7. Crfo^t +• Z £ co 4-b 6 4 " 5 - 4 - 64-( A 3 . \ i o ) With the s i m p l i f i e d expression for v c (see Equation A3.92) ^ s ^ t + p ) + 4 - ^ ^ s - ^ t + e ; + ^ , and from Equation A3.104: because cos 8 ^ = 1 . The expansion for y i s of the form (A*.!!-*) so that , from Equations A3.100 and A3.103, we can w r i t e : +9) = ^(Zo^o cos 9 + ^ ^ [ ^ > ^ + / V > + / V ^ > j c o s - B 0 s ^ t (03.114-) 280 Equation A 3 . I l l can now be expanded as V ^ 1 - 2 ^ 0 - c o 5 z t ) + t+fiofa .cos 9D s ^ t + ^ A3.32 + 6 4 / 5 ^ , 0 -CP52t) 4. ^ | / 3 a ^ , + / } , ^ C P s P o f f ^ t + 4 £' When the secular term in Equation 4.61 i s neglected (see the discussion on page A3.27 ) we have: $ -and consequently = 2 tZb - 4-s;~b (A 3. H o ) A f t e r subs t i tu t ing for 3 , sin(.t+6) and y from Equations A3.100, A3.103 and A3.113, the expansion of 2 v 5 v g can be w r i t t e n : 1L s ^ l b - 4 j ; ^ b ^ - /4 0tos9 o | 4 - - J l co S b -4-oo"2.fc +_Mco.?3 + e 5] 281 4- £ A3.33 (A l. u?) When evaluating the quant i t ies Vg and 2VgVg, the reduced form of Vg(t) and Vg('t) must be used (see page A3.27). Apart from t h i s considerat ion, the remaining terms in Equation A3.94 are r e a d i l y evaluated, and from Equations 4.61, A3.100, A3.103 and 4.84 we obta in : "4S - u_ c o t - ScsaZfc + 11 c o 3 t - i i i c » 4 t CA3-H8) l V $ V g •/*,r«-/*0 Cs>l (A3.\n) For the product u^v^: U.s\/S = £ 2.^-^SecoCt+p) J j 2^}<cCk+P) + (A3.no ) which can be developed as From Equations A3.102 and A3.103: 282 A3.34 and also 2±y e*,tfo + e\*,^r, + ^ ° < 3 ^ ° ] J + (*3-*3) (see Equations A3.98 and A3.113). From Equations A3.103 and A3.104 for sin(t+9) and cos(t+9) we obtain s u t ( t + 8>c©s(t + e ) = j . s * i b 4- e39.c^Zt + O f e O (/ij a t ) and therefore, using Equation A3.101: - e * ^ / s , * " + 2 /*<»/V ^ b - e 3 £ z s s-zfc + 2 ^ 0 X 0 3 <*>zbJ + o C f e O The f i n a l term in Equation A3.121 can be wri t ten - e 5 where 3 , cos(t+9) and y are given in Equations A3.100, A3.104 and A3.113. When the reduced forms of v 5 , v^ and v g are used, the remain-ing terms i n Equation A3.95 take the form: 283 A3.35 n>Dc<nQo | II 5«t - 2s:«.2b + i i sc«.3t^ 1 i ifc It J (A 3.ix?) Jkrab - GoJU: -1 '4*1>4 £7 s~b - 4 2 sCv2.fc + W *.W3b -8 s e J i s;~4-b j J (A*.1*1) L 3: sc~2.b +j£ $^ .A-b 2. M --+- O ^ ) (A*.r&o) [ I T bk bit J ' I 32. 3 . M - ] ( A 3 . 13-1) 2 8 ^ We are now in a pos i t ion to assemble the various terms i n A3.36 2 2 Equations A3.92, A3.93 and A3.94. The r e s u l t i n g expansions for e u , 2 2 2 e v and e uv are given below. 4of,'1 -- Z p , ( l - f O ^ ( £ I - 3^  cob + toJ-t-.1 c^ >3b ^ + 4 p , ( l - fO<*-, f ^b-<*>2.fc --t — o t t o i , t- •*- t» <"-U — Cao >i 13 Z f S ' l ~ ^ t o 5 0 | l - 3 + c^Zt- - ± to3b ^ t- e 285 A3.37 3 1 Z p / l - f O p>0cer,9c> ^  4 - i i . c*>b - 4 felt -t- JL c o 3 b ^ + 2 ^ , 0 . - ^ 0 j J i si«.ib - 4 s « .b? (A3.»33) Ca b - S c < o 2 . t -t- j | c o 3 f c -2. I?-] Co 4 b 7 13 siwZb — 4 s:»vt 3 - 2 / * , ( l - / 0 fxe*>90 £ A- -JL <vot - 4 <*>ab +. \± ^ 3 b - ^ - 3_ Xb - is oo4b^ 2- J6 J -4- . 0 ^ * ) 286 (i0 5 ; ^ t .+ poyo coPoCeot A3.38 + 6 + f i ^ - f O ^ 11 ««-Zb-gsiwt ^ - ^5,coP 0| Jl -1«.~Zb -t- i i s^n ^  •+ plxC\-p7_')L 5 7JJ- s;«.b - 43 s«2b +• xj $tw3b - Ji s;~4t"£ L s & s xt- J ( h 4 . i ' *4 ) b -2«;_x.b 4- Ji s^-ib ^ - ^ 5 Zcob- co l t - i + /3^<ic90 ^ s ^ b - Z s ^ Z b + 5Uv3b j L I- 3Z b^ t- ^ i - 4- *>*- 64- J 287 A3.39 The nonlinear functions and can now be evaluated for a s p e c i f i c order i n e when they are required. Af ter t h i s necessary, but lengthy, d igress ion , we return to the analysis of Equations A3.22 and A3.23 for u ^ and v ^ . From Appendix I. Equations A l . 8 and A l . 9 : CA3.I3S) ^ OCi> ( f l 3 . ' 3 0 Now s i m i l a r terms also appear i n the expressions for E-|2» F 1 2 ' E 13 a n d F 1 3 (see pages A1.4 to 1 A1 .6 ) , S o i t w i l l be expedient to evaluate , i n a d d i t i o n , the contr ibut ions 0(e) and 0 (e ) from Equations A3.135 and A3.136. We have, from Equations 4 .25, A3.98 and A3.104: U.,- = £. 2erf, - CA3-I3?") (A3.I3S) Note, i n c i d e n t a l l y , that ' the complete form of v g ( t ) i s used here, in contrast to the reduced version given by Equation A3.92. A f t e r some elementary manipulation: 288 A3.40 - € r ^ot , - + t o t , c o z t ~ pi <^>90 ^ 1 px^e* I ] + 6 3^ ,«<o^ ^ ] V $ ( l - 3 c o Z k ) = ^ 0 4 o P 0 ^ 5 j ^ b - 3 s ; ^ 3 b " ^ + C v - 3 c « X b ' ) + 6 s;«.Xb = - 2 - t i o ^ 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 fo l lowing form: I t 290 A3.42 General expressions for the q u a n t i t i e s - -2 £jf-s - ^fs and . Ibtbv bV J -2h>j£s + a_u5 a r e given i n Equations A3.28 and A3.29, but at t h i s stage LdbSV at J of the development Equations A3.30 and A3.31 are no longer adequate. We now know that | | = 0(e 5), § £ = 0(e 5) , | * + 0(e 5) , | | = 0(e3) and 6 = 6Q + 0(e 3). With t h i s value of 8, equations A3.14 and A3.15 can be revised so that a^'C e*9y A- + ) = e% 0 3 + ' e * ^ +• fes0s oC**> CA3.IV?) cos C -*• ••• • ) = 1 + O C £ « ) . (A?.UtS) Consequently Equations A3.16 and A3.17 take the fo l lowing form: since s i n e Q = 0 (see Equation 4 .78) , and we can wr i te Hv ' ar av t j r ^ J (fl3.i5i ) - (i.h*2* - AcQ+Zjj -fi,9* 3£J \ + O f t ' ) (. "^r • a v air J ' ar ' aV [_ at av j t- atr ar / av 5 u . aV ' ' <>vj * 6 * j / 3 „ < ^ * . + - • • . ^12? \to60+ , .(fl3.'J2"> L a a*c 291 A3.43 where Equations A3.151 and A3.152 are the revised form of Equations A3.30 and A3.31. From Equations A3.28 and A3.29, and using the above expressions: -% d U5 — iJs - 6b ^ O i y -Z L ab«?r iv {' ix i r 1 i v S .b - 2t*>®0df>s Cost a r 2V In addi t ion to the orderre la t ionships given on page A3.42. we have 3V= 0(e 5), = 0(e 5), £r = 0(e5) and = 0(e 5). Consequen-tly' at" a t 1 a t * t l y (from Equations A3.34 and A3.35) fu* and 3V S are both 0(e ), a t * at 2 -so that From the expansions of db and [ * a V L iv-_ given i n Equations A3.69 and A3.70, and with 8 2 = 0 (see Equation A3.83) , i t fo l lows that 1 < ^ ^b J 0 / , V > L ' 1 <h dVj +- c)u 5 ^ b x 5b - e r a 2 9 2 A3.44 With the order re la t ionships given immediately above, together with those on page A3.42, none of the expressions defined by Equations A3.74 to A3.76 has any contr ibut ion at 0(e) or 0(e ) i n Equations A3.22 and A3.23. There i s , s i m i l a r l y , no contr ibut ion from terms i n v o l v i n g the der ivat ives 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 for u g ( t ) and V g ( t ) , we o b t a i n if- bb Zi + _5_ ts^fc + I3S Co 3b £ 64- 64-if- bb f O ^ j" ~3_ s;^k + WHS s»\v3.b A l s o , from Equations 4.25 and 4.26 for Ug and v^: = p> c^> ( b + 60 <)b^  (A3. 16 o) .1-a_v5 - 2/b J<«-(b + P ") and therefore , using the expansions of sine and cose i n Equations A3.149 and A3.150: b_U$ btx cos b aS/s ^ - 2 c o 0 d J * /s 0 +• e* A x I s;^.b + o (A.3.lfrl) (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 ^ a^ at? 3 5 2 8 o - / o L at Cft3.ifc'5). I av a IT av j 3s. 29k . A3.46 <*3 pxfb0c^90 - us / i l0 + / t | . W - / O X 5C«. 3 t When ctg s a t i s f i e s Equation 4.82, the two terms 6( 1+jO *~°i-3 and - . i l / t t C l + w,)^ ( i - /x .^) 2 - i n Equation A3.164 cancel (see also page A3.15 4 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 involv ing t i s to appear i n v ^ ( t ) , then from Equation A3.9 : 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 der iva t ion of Equation A3.89. To e l iminate t s i n t and tcost 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 condit ion for uniform v a l i d i t y i n x unless 9 ^ i s zero, so we have 6> 3 - o Chz.no) - 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 i n Equation A3.169 i s now 1 oX zero , and a f t e r some manipulation we can wr i te t O * £ >4j (i+p.r*- +• \->s ? = o . (A3. p / \ z Note that cos 9 N = 1, so that a cosBa = H u, c ' ~ (see Equation u ' 1 6 0 + 4.81) . A l l the terms in Equation A3.172, with the exception at present of sL©5 , are constant (see Equations A3.50 and A3.86) , and consequently d©5 - constant (A3.t?3) 296 I f we also assume that and 6^ are constant: A3.48 sec 9, (ft 3.174 ) (see Equations A3.211 and A3.244). From Equation A3.9 , one secular 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)coiB0 - 2 ^ , 0 - ^ ) i t fol lows that (A3.(?S) (see Equation A3.53) , so that the secular contr ibut ion .can be wr i t ten 297 A3.49 The remaining secular terms in v-]](t) r e s u l t from quant i t ies i n Equation A3.165 which involve s in2t and s in3t (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 in2t in 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 6 (3tag) i n v 5 ( t ) : (A 3.i ? 0 As u s u a l , we take -which leads to the fo l lowing value of SfjL,( | _ f O ^ " [ l < ? + ^ 3 ^ , ] - C A 3 . I T 8 ) The functions Ug(t) and Vg(t) can now be evaluated, since the values of 0-j, 6 2 and 8 3 are known. Equations A3.42 and A3.43 s i m p l i f y considerably, and we then have 298 A3.50 Non-homogeneous solut ions corresponding to these functions can r e a d i l y be derived from Equations A3.10 and A3.11, and when the secular term i s included ( c f . page A3.17) u g ( t ) and v g ( t ) take the fo l lowing form: + 6± vsb 6 - \± Cos, 2-k 6 (A3.(61 > 3 £ 2 s* Xb + XI b I (A3 . I S O Next we consider the solut ions of Equations A3.24 and A3.25 for u.j2 and v - ^ . From Equations A1.8 and A1.9 i n Appendix I : 6A-'7-S (to3tr + 3|5 ^ S f c -i » 9 1-2.8 (A3-lS3^ ' i-JS si^t + IOS S ; ^ 3 b -+ ?rS <cHSb 64- is-S ra-S CA3.I24-) Solutions for Ug(t) and Vg(t) are given on page 4.15, and a f t e r some manipulation we obta in : H,A 3<*>Zb) - 3V. <^.xt 6 • CA3-I85) ' 1 /6 i6 4-2 9 9 •A C I - "3 Ccr> 2 -0 - 3 u . , S<«-2b 6 «» A3.51 8 /6. 4-( A 3 . I 8 6 ) From Equations A3.139 to A3.142, the contr ibut ions involv ing u^ and v^ can be w r i t t e n : 2 b ^ - 3 v , -2.1 } O f t ) (f\3.lg?) £ - il c ^ t + £ < ^ a b +3<o>3t + 6 2-I S _ t S i - 2-b \ J ^, Cl- 3 c n i t l (A3.I88) + -M-.CI- / O 2. ? Equations A3.183 and A3.184 can now be evaluated, and we have: 96 4 3-2. - 2 p . ^ i Cfl3.i81) 7 £ c o b + ]75 cr>3t 64- t 3 . 8 + 3 i £ <v» S b l>8 i 5 t i t + ioS s ^ b + 3jj> b 64/ i a S U S (A3.l<*o) J6 3 Z rV2Tr Cl- 3 c « 2 . t ) 2. ' Note the cance l la t ion of secular terms i n v o l v i n g t s i n 2 t and tcos2t , which appear i n Equations A3.185 to A3.188. A f t e r some algebraic manipulation the nonlinear functions can be 300 A3.52 arranged i n the fo l lowing form: II 5x (A3. Ill") [ sew Zt Zfc (see Equations A3.132 to A3.134). The quant i t ies - z f ^ j f s - "1 and - 2 T i*y> + ^ 5 * 1 |_atav J oCfc6) L a t a * a r J oCe6) 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 oCe6) + Zc*o©( I ' br dv J 301 A3.53 [ x -a v/g + }tdx Tx SA ,aP t +/>,iJ£f + ^ ^ 7 S £ ^c L >v ' dx bxr S dx (ft 1 . I S O because, from Equation A3.170, 6 ^ = 0. At present the der ivat ives of a , B and y with respect to T are a l l 0(e 6), |^-=0(e 4), ^9 = 0(e6) and 6 = 9 n + 0 ( e 4 ) . From Equations A3.34 and A3.35 i t fol lows that $j±_s and ^y_s are 0(e 6), and consequently dx* _ =. o OCeO With 9 2 = 9 3 = ° ' E < l u a t i ° n s A3.69 and A3.70 s i m p l i f y considerably, and we obta in : L at1- at J (p^*) OCe*) COS None of the quant i t ies defined i n Equations A3.74 to A3.76 2 3 contribute 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 for u g ( t ) and v g ( t ) : 302 at"1 '<Jb . L 6 3 From Equations A3.162 and A3.163: c o f i ( , c o t •2/3, eoP 0 *-.t A3.54 (A3.119) • (A3. I l l ) (A3. 2 O 0 ) (A3. From Equations 4.60 and 4.61 for Ug(t) and Vg(t) = p i C ' - / o F 4 t<olt - 2<uot ] C » - 4 - s u v t - J i ^ Z b J and therefore , using Equation A3.53 f o r B^COSOQ: aV at* cn It (As. XOTL) (A3.-3.03) (A3- ze>4-) (AS.ZoS) We are now in 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: *>r 768 1 a r J ( ' - ^ O J 303 + c o t £' a * " ' a r ' a v J A3.55 6^- 3 • - OS -j**.") fco3fc 13.8 2- Sott •+ sovcb r ** a*p" <?v J £ 4 -aV C - r o ( j - r o 4-17-S 3Z Secular terms involv ing t are el iminated from v - ^ t ) i f (A1.208) so that equation A3.178 for a g i s v a l i d . To remove terms containing t s i n t and tcost from u ^ and v - ^ : 3014. A3.56 *- ' a i r d~c J From Equation A3.209, i f 9 4 i s not zero 3g(x) v io la tes the condi t ion for uniform v a l i d i t y i n x, and therefore ^4 - ° ( A - i . a i t ) Now when 6 4 = 0, we obtain from Equation A3.85: i n which case Equation A3.210 can be wr i t t en 'A-. Iff. / « , 0 - / O -+ '£L K . ^ f O ^ - j O =o 6 4 V From Equations A3.84 and A3.173 both 3. and 3_f5 are constant, so that 4 ar 305 SSfc = constant . at? A3.57 (A3.2IS } I f we now assume that e 5 and e g are constant, then from Equation A3.214: 6M + 105 ( i - - fO~ b —. s i« ( n- JJ. , ) sec 9r (A3.lib) The quantity - 3 l ' av ' J 64- 3 - z a-rr ? 6 g which generates a secular term i n v - j 2 ( t ) , can be s i m p l i f i e d using Equation A3.214, so that the secular contr ibut ion takes the form [ 3 2 - 3 S 4 - . 0 . - / 0 ~ S4- fa. Addit ional 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 in4t and s in5t (see Equation A3.11) . The combination of these secular quant i t ies must e l iminate the term -e^(3tay) i n Vg ( t ) , and therefore : 4 v* (A-s .ai? ) We take, as before, 306 A3.58 and i f i s assumed to be constant (see Equation A3.240), from Equation A3.217 i t fo l lows that i X f 0+|<t) + ££ Yi + ix- 3 £ 64 -( A 3 . 2 l < 0 Now that the values of ^ a n d 6^ are known, we are i n a pos i t ion to evaluate u ^ ( t ) and v ^ ( t ) . Equations A3.77 and A3.78 can be wr i t ten i n the reduced form: <Vo2-t +• 3 f p,(i~pj) Cr>A-b ( A 3 . 2 X O ) 16 0 - p O ^ 8 TT^o ' 6 and from Equations A3.8 to A3.11 u 1 Q ( t ) and v-|g(t) can therefore be derived as: flt(l-ftj.^ 37 + 6 7 S ^t, c o b L 8 s - ia-Ci+jJ.) J 3 a 307 A3.59 - i I f . + X-zS fi, fr3 s^A-b L 6 4 - S ixC l - t - u , ) J - 3 ( b - s^.b) 0-/**) - 4-S ^ 0 ( V c o b ) ( f l 5 . « - 3 ) Note that secular terms involv ing t are el iminated in the complete so lu t ion for ev(t) when s a t i s f i e s Equation A3.89. We consider , f i n a l l y , the s o l u t i o n of Equations A3.26 and A3.27 for u-j2 and v-jg. The functions Uy(t) and v^(t) are both zero, so that from Equations A1.8 and A1.9 we have ' I S - u , ( l - f O | f 1 <»k + l l c « 3 b I - V 5 J 3_ s ; I ' L « e J L * s .vb +• *5 5C^.3b 4-s ^ 2 - b ? ( A 3 , aa-S) Now U s r - p>0etr>90 C o b • + O f t ) N/& = 2y3„ Co0o s~b + y0 + oCe^> .', 3 0 8 A3.60 and consequently: 1 tot +• to?t 3_ s;~b + if 5J«.3b 4 4 . 3 5 « t t if 4 4-sc^3b 0 Csp >x0 if + 6 co 3-t - if <<o4k 8 « ( A 3 . 1 Z 6 ) [ & 9 _ -y0 [" 2. c ^ b - i f c*>3b [_ A- 4 - V c — 1 tot - l£ < o 3 t 4 4-e>Ci> I? s i^Zfc - if j;^4k 8 ( A 3 . Z 2 > ) Because 3 2 = 0 (see'Equation A3.213) Equations A3.139 to A3.142 s i m p l i f y , and f o r E ^ ' a n d 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 27 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 contr ibut ion to Equations A3.26 and A3.27 can be wr i t ten as: 3x ' 309 A3.61 . 4-CA3.230) [ i l i t 64 8 For the expressions -2 | d "-s - ^ s l • |_ d t a r J we have, using Equations A3.151 and A3.152: CA3 . 23\^) and -2 }L?S + ^-s , -2. it7r ar J QCtT) ix >x ( A 3 . 2.32.) + X t)/3 ? ceoPp s ^ t + 2&90f A0bB? 1-/3,^6 "I ^ s t L >k3 ; t a t r 2 a*,. atr ( A - i .133) 2 CoP„ cost , •tr 310 A3.62 since $ 2 = 9 4 = 0 The der ivat ives of a , 8 and y with respect to T a re , at t h i s stage of the a n a l y s i s , a l l 0 ( e 7 ) , | ^ = 0 ( e 5 ) , $9 = 0(e 7 ) and 6 = e_ + 0( (see Equations A3.208, A3.211, A3.212, A3.215 and A3.218). Consequently z v 5 the quant i t ies 3_!f5 and a_v> do not contribute at 0(e ) to Equations It*- bX*-A3.26 and A3.27. I t i s necessary to extend equations A3.69 and A3.70 to include terms 0 ( e 5 ) , but as 9^  = = 9g = = 0 t h i s presents no d i f f i c u l t y , -and we obta in : I a t ' * a t j a_y5 + h±s c Laf- at J There i s no c o n t r i b u t i o n , e i ther at 0(e 4 ) or 0 ( e 3 ) , from the expressions 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 - a * , at' a t /- " , (<-r 3- V-3.Z5 p , ? tost + f - p., ? coj 2.t + 2il cos4t -64-S T3 + 6 ? £ ^ _ 7 L 6 4 - S - ia-Ci^p.) J + 3^ [ X - .e^ -b 1 -f- -45 j< , 6 - j O 2o s ^ t (A3.a34) 311 A3.63 - - 3 ^ s u l : - 4-5 ^ . Q - f Q y e t o 5 1 CA3.23?-) + ^ , 0 - / 0 * f .^ 4- za-g'/x, ? - 5 ^ b - ' / i ! > + 16 The non-homogeneous terms in Equations A3.26 and A3.27, which we can now assemble, take the fo l lowing form: - 6 t ^ , . ' H . _ 3 _ f / 4 . / i 3 + ^ 3 * " ^ ] - + p / ' - f O / * }Xr a v f ' l - ^ , ^ 64-8. •+ «o b 1 J 3 2 •'+11 (*,yx^« r + 75 su-3fc •+ cos4b 31 *5 j« . & - f O ' / v * 0 . - + i i i p / ' + f O ' ^ ' - r O 3 64- 3 * CR3.2.38) Z 3o£> -t-S^b (.' ar <ar J 312 A3.64 - <vob - s;».zb - , * 64 L 4- g 2 I f no term involv ing t i s to appear i n v 1 3 ( t ) , then from Equation A3.9: ^ ? = O , Cfl3.Z4o^ a r and therefore Equation A3.219 i s v a l i d . Terms containing t s i n t and *cost are el iminated from u-j^ and v - ^ when: a* it t ar ar Although i t i s not required in the present a n a l y s i s , i n pract ice we must take YQ = 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 zero, 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 83 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 . From Equation A3.215 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, then sec 9 Q . -(A3.24-?-) o The secular term i n v 1 3 ( t ) must e l iminate the quanti ty -e (3ta g ) i n v 5 ( t ) , and from Equations A3.9 and A3.11 we then have: + W } - C j r . ^ * ^ . ^ ] . - ^ i a > v 3 * 8 ( A * . * * * ) By the usual argument: 3114-A 3 . 6 6 and, assuming ctg i s constant, from Equations 4 . 8 1 and A 3 . 1 7 4 for 3Q and $ 2 we obta in : 0+ j*,)*" ( l~ / O ifj + 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 comparison with e a r l i e r resul t s (see Equations A 3 . 8 9 , A 3 . 1 7 8 and A 3 . 2 1 9 ) . This completes the general analys is of 3-body motion, although solut ions for ^ ' ( t ) and r ^ U ) derived from these resul ts are given in Chapter 4 . 315 A4.1 APPENDIX IV — VARIATIONAL ORBITS IN THE RESTRICTED PROBLEM We are concerned here with the homogeneous and non-homogeneous solut ions 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 equi l ibr ium points of the r e s t r i c t e d problem. For the non-homogeneous solut ions 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 equi la tera l t r i a n g l e po in ts . The present a n a l y s i s , however, i s l i m i t e d to the inves t iga t ion of motion near the equi la tera l t r i a n g l e po ints , so non-homogeneous solut ions for the c o l l i n e a r case are omitted. A4.1 The Homogeneous Solut ion I f we take x =x-j, x = x 2 , y = x 3 and y = x^, then (neglecting the non-homogeneous terms) equations 5.23 and 5.24 can be wr i t t en in the form 3c = A (A 4. 0 Here x = (x-j, x 2 , x^, x^)^ , and the matrix A i s given by A o "°. o •o. I o o -z o 0 o 2. i O ( A4. % ) 316 Taking the Laplace transform of Equation A4.1 we obtain A4.2 (A4. 3) where I denotes the unit matr ix . From Equation A4.2: and i f we now define then Equation A4.4 can be wri t ten deb ( s i - A ) = r2- + 2-X, r + x z (A4.6) where From Equation A4.6 we therefore - have d e b C s l - f l ) = j s V X , + A , - ( A * - \ ) ^ (A4.7) ( A 4 . g ) (A4.<0 A f t e r some manipulation the fo l lowing r e s u l t can be obtained: 317 A4.3 ( s i - A ) — i 3K C <Lefc ( s I - A ) ( A 4 . lo^) where C denotes the matrix: 2s-u, 20,-o^s s1-*- o. Note that the quant i t ies and depend on the p a r t i c u l a r values of £^ and ( cf .Equations 5.20, 5.21 and 5.22 on page 5.6 ). « • - - - • - ... — J . J i . -I ... - .!. I. - - ~ . . J T - J . _ T ^ _ ... _ T _ ~> — — ,~ J - — n c HUH ICOOI u u i u o i , c i i o i u i i u O o n e c i { u i IUICIUI _ C i i c i n y i c i m , o and 1_5 (see Figure 3-2 on page 3 .18) , for which L7- L2> - 1 Note, a lso, that + 3^ = 1 5.21 and 5.22: 2 , at L Y (_A4. iz) y r / 2 at L C ( 0 4 . 1 4 - ) In th i s case we have, from Equations 5.20, o, - — ' - K - k, z 318 at L 4 : and at L^: A4.4 3L 2. From Equations A4.7 and A4.8 i t fo l lows that \ = 1 ' + k0 + K (A4 .n) V = (l-^o)'' - * k / l - k 0 ) + 3 1 ^ , 0 - / 0 (A4.ZO) and consequently \ x - \ x = A - C ^ + O + k,7" - 3 k ^ J i - r , _ ) . ( A 4 . 2 0 Note that Equations A4.19 and A4.20 are v a l i d both at and L g . The character of the homogeneous so lut ion i s determined by Equation A4.20, and bounded motion only occurs , i n general , when 4 ( k 0 - r k , ) + k,2- - S k ^ j ^ f i ' / O > O . r«4.ix) I t i s possible f o r bounded motion to take place when Equation A4.21 i s not s a t i s f i e d , but only for s p e c i f i c i n i t i a l condi t ions . We now r e s t r i c t the so lu t ion fur ther to those cases f o r which Equation A4.21 i s v a l i d , 319 A4.5 which implies that > i _L - 4- C i<e + k, > - j "5 k o r , a l t e r n a t i v e l y < j_ -2-- J 11-"7 J-3 k since u 2 ^ VJ3 requires that u 2 ^ 0 - 5 - 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 wr i t ten * i t 1 - (r,)'1 19 which i s i n agreement with the value given i n Szebehely [1 ] . Taking the inverse Laplace transform of Equation A4.3 we o b t a i n , using Equation A 4 . l l . : • Cb^ = si*. u>< b Zo^Tcto^ + (oi^-O^ £Co"} + 20^ LJCO-} + 0 ^ <j<o) - C 0 5 cos fc 320 + Cos k A4.6 i(fc) - - s ^ i ^ s b 2 0 , 3t£o"> - o ^ i t o ) 4- 2 0 ^ 3 ( 0 " ) -(- ( o , - ^ 7 - ^ y ( o ) to . + COS o><t + S i t t J u t 2 0 , =c(o") - 0^ i C o " ^ + ZOa. LJCO") + ( o , - ^ - 0 ) 3 ( 0 ) - cos ojt (A4.1?) where I t i s more convenient, however, to express x( t ) and y ( t ) i n the fo l lowing form: (ft4 .ZS) C A 4 . 3 o ) ( 0 4 . 3 1 ) where a , b , 9 and <j> are a r b i t r a r y constants. The remaining quant i t i es Q ,Q - , 6* and <j>* are defined below: 321 A4.7 Q u = [ 4-oJ- -»- 0? ] " (A4 .33 ) ^ (A4.34) Note the symmetry between expressions associated with the short -per iod (u>s) and long-period (a^) terms which i s revealed by th i s formulat ion. 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 wr i t ten £ - 2 ^ + o , x + -O^tj = E * ( t ) ( f t 4 .36 > ) y + Z i + t o 3 ^ » F * C t ) . ( 4 4 . 3 ? ) I f we se lec t the vector B so that B = (0, E * , 0, F * ) T , then Equations A4.36 and A4.37 can be expressed as S = f\5c + 5 , CA4 - 3 6 ) 322 A4.8 . where the matrix A i s defined i n Equation A4.2 . Taking the Laplace transform of Equation A4.38: 5L( S^ = ( s I - A ) _ , S C o ) + ( s l - f l ) " ' ZCs) ( f t * . 3*0 and the non-homogeneous so lut ion 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 sinut and cosoot, three d i s t i n c t cases are poss ib le : 1. w s f co f u L ; 2. w = oos ; 3. oo = co^  . These are considered separately i n the analysis that f o l l o w s . A4.2.1 Non-resonant case When cos f w f co L , E*(t) and F*(t) can be wr i t ten i n the general form £ * ( b ) = C<„ +. Ct, Swvtofc- + o L , . cos o b '. . ( f l + * * o ) F * C f ) = / S 0 -t / S , s f ^ t J b + / J , . cx>5 t J t , . ( A«r. <*i) and we therefore have 2» s H i O 1 - S ^ + O - * -323 A4.9 B(s) i s now known (from Equations A4.42 and A4.43) , and the non-homogeneous so lut ion may be.derived from Equations A4.10 and A 4 . l l as the inverse transform of ( s ! - A ) _ 1 B ( s ) . A f t e r some manipulation we obtain: 2 A UJJ- ( toot, »2|Q + •oCft.Q.,.-*^) + 2/3, + cos o t b ^ ( CO*-— cx>s tob (A*.44-) — SiVv u><- b 32k A4.10 — Cos t->jt-— 2-ot, — StK-0>fc - to$ cob I ( coL- O -+ c i x O x -V 2-coot, ( w s * - W " 1 ) A4.2.2 Resonances involv ing short -per iod terms For the second case, when co = to , E*(t) and F*(t) can be s expressed i n the fo l lowing general form: F * C t ) = ' 3 " . * * . o , b . + S", w s c j j b t ( A 4 . A - ? ) 325 A 4 . l l . so that ( A4. The corresponding non-homogeneous so lut ion of Equation A4.38 i s given below. MH — Siw-lO.fc to. - cos to. b t Si«- to-b Z t o s ( t o ^ - c o j O - t cos (Os b 4- 2^,05*1 (A4, 5-o) 326 A4.12 NH COS t CoS«0, t tcc-5 ^  t (fi4.Sl) A4.2.3 Resonances involv ing long-period terms When to = to^  we have: (A4.Sz) (A4.53) and therefore 327 A4.13 The non-homogeneous solut ions 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 s and o ^ . Note, however, that i n th i s case (to - t v s change. We then obtain ; ? 2 2 2 a£ o£) transforms to - ( t o s - w L ) a f t e r the i n t e r - cos t^k 0 ^ e VZJ, 328 A4.14 Cb) = " sew. cjj b-u^ L S^C^-O + J*-y«>l - 2jr, V - cos <os 11 C J J ^ - « Q + °j-}f\ V ° T ° ' -t + t cos to. b 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 der ivat ive terms r e s p e c t i v e l y , are given below i n Sections A5.1 and A5.2 . In the f i n a l sect ion we continue the analysis of Chapter 5 to derive solut ions for eu(t) and ev(t) that are uniformly v a l i d when t = 0(e~^) and which define the pos i t ion of p^ to wi th in a 8 constant error 0(e ). A5.1 Expansion of the Nonlinear Terms For the function \rd- | we have, using Equations 5.32 and 5.33: Uniformly v a l i d expressions for the motion of p 2 and p 3 are given i n Equations 4.101 to 4.104, so that 1. r 330 A5.2 I f Equations A5.2 and A5.3 are rewritten in the fo l lowing form: then we obtain Now because of the loca t ion of and (the equi la te ra l t r i a n g l e p o i n t s ) : - C> ~ - • + = Cfx+Sj' + I?'' = • .. ( A S . * ) In t h i s case Equations A5.6 and A5.7 can be wr i t ten o r , more s imply: 331 A5.3 4 3 where . From equations A 5 . l l and A5.12 we then have the fo l lowing general ser ies expansions of ' and I anrl I 3 3 r 4 l | r P k o + k, ^ -t- Lrt g v + . • • (A*.iO ( cf . page 5.4) , where the constant c o e f f i c i e n t s depend on the type of expansion i n i ( for i = 2,3) as: use. The expressions - £) and (n. - n) can be 'wri t ten s " ^ 1 * 0 . \'.(AMI) ^ - 5 * 0 . - * i (AS.H) = - AO ( A S . * * ) 332 A5.4 and consequently the nonlinear terms i n Equations 5.36 and 5.37 take the form: + k. + k. ( f t S . Z l ) where we have made use of the condit ion i - . (A5.0.3 ) which i s v a l i d at and L^. 3 3 3 A5.5 Equations A5.21 and A5.22 cannot, unfortunately , be used i n th i s compact form, and a development i n terms of e i s now necessary. The expressions for , T^, A-| and A 2 are more r e a d i l y handled i f we take IO / • — J «--S A » « e u - + C 1 e ^ O ^ + O C e " ) ( A s . * * ) where e^(t) and f^( t ) 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^) (see the discussion on pages 5.13 and 5.14),-we obtain from Equations A5.24 to A5.29: P,1 - c V - - 2 - f c V e s + fe'°cs* + OCV') (A5-3o) 33k A5.6 e V + L I -L I --t- O d e " ) (A«.3S). 0 - / O 0 - p * ^ (A5.35> A l s o , i f quant i t ies ( H e 1 1 ) are neglected, the fo l lowing approximations are j u s t i f i e d : - A. + , —3-CA5.-56) ( A S . I f i ' ) ( A S - A o ) ( A S . 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-^ uC P,P,_-A,^) + ( 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. 11 2 2 Note that , when quant i t ies 0(e ) are neglected, the terms f r ^ , g , 2 2 f and g A 2 make no contr ibut ion to Equations A5.21 and A5.22. A f t e r some rearrangement, the nonlinear functions can be wr i t ten i n the form given below. CAS.4t) + < ^ 2 ^ * v ( k , + O + P - g ^ ^ l ^ O + v ^ k , ^ ^ ) ^ i - ^ O L *• • • 3 3 6 A5.8 + fe + 6* u.k , *f u 6- £ / 0 V | k 0 t 3k, | CA5.47) 0 - . Note that , although the value of ri|_ depends on the choice of Lagrange 2 point , n£ = 3/4 at both L^ and Lg. This value has been subst i tuted 2 for n£ in Equations A5.46 and A5.47. The d i v i s o r ( l - y 2 ) associated with terms involv ing e 1 0 could , 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 Equations 4.101 and 4.102: ^ s « • ' l? j O * S-K-b , (AS. 4--0 8 (*!+ and therefore as y , 1 the terms i n question tend to zero. 337 A5.9 A5.2 Asymptotic Expansion of the Derivat ive Terms From Equations 5.58 and 5.59, the homogeneous solut ions for u^( t ,T) and Vg ( t ,x) are defined by: and therefore : av ^ t r ^ U-s = — u>s y bo. s^ »v Titx I bx S - u>u / £W s:-vEtouk + ^ ] + bj> cos E^ut+-</>] Z I bX bX J ( A 5 . S 3 ) " 5 ' iP'VLf = COS (BS --5-4) Corresponding resul ts for the der ivat ives of Vg ( t ,x) can be obtained by inspection from Equations A5.52 to A5.54. 3 3 8 A5.10 Expansions for sine and cose are given in Equations A3.16 and A3.17 (see Appendix I I I , page A3.6 ) , and using <Ost cos 9 + coso$b si^9 (As - , ss) C o s ( ^ 5 b + 9 ) = Co5u>jt « > 5 # - S ^ « O s t S<^-0 (AS. ? 6 ) we obtain -s;^o 5k |Vv.9 e + feD.cosB* + t 1 ^ x c o s 0 o - ©,1*H&OJ + • - J (As.*8) Equivalent expressions for sin[w^t + <j>] and cos[o^t + <j>] fo l low from these resul t s by symmetry. From Equations 5.62 to 5.65 the functions , , a-^- and can be wr i t ten as: ax dv ' $v 3 3 9 A 5 . l l a.b9_ bxr [ a v a v a t -oCfeO ( A S . 61) bbj> bX? a r bV d~c 0 C e 3 ) and using Equations A5.57 and A5.58 we now have 3a co* ( W j k 0 ) = / co5«J s b cos 0O - si^o5t- sc*-0o 7 at I J + e l cos (p s t + 9a) L a i ? at? } •/ r ^ 1 L ^ z- a r j cos C to 5 b +- D 0 ) [ } + 0 ( V ) CA^.65) di> cos(toufc+^>) - atD S cosOLt cos 6 - sc* oub si*. d> 1 dY « 1 ' J L a*c ° 1 d v ° J 31+0 A5.12 { bh, --J3-bV cos C u>ut + C^0) Co. t + c O + o ( e 0 e)a si>v( u s t + G 0 ) = ba.0 S si*.u>sb c*z>0o + co5 0 5 t s ^ _ E ) 0 7 dX bV I J J Sa, SC«- (u> 51 + 6 0 ) +• G, cosCcOst + 0 O ) "I I L bx i- bxr j 9, ^ce, + G^cte.,, cos(u>s b+ 8 0 ) } ( A S . 6 5 ) } 1 a r  xTx cos ( ojc + <j>0) + O l V ) ( A 5 . ^ ) A5.13 o. cosC u>5t +9) - a0c>Qo I cos u>sbcos &0 - $i^u>st s<.v. 9Q \ bX bX L J a^bQo -f- a 0 ^ 6 | Cos( u)sl + 6>0) - ao0, 3P0 £ <Osfc + 0O) I J L i f d v 3- \ l-> r> rrtC ( o h + r£0 — ' - I- ( oX cos ((j utr -f </>0) - hoj>x~c\i>0 s^(o i_t+^> &) at? + 6. 1 J l I r <?r a r 3 }xr d X ^ J d9 5 ^ ( i o 5 t + © ) ^ a 0 d9 0 I s ^ v . ^ s b cos 90 + tosu>5\:si^Q0 "I 3k2 +• e C ost 4-9 0) a o 0 , COS £ to51 + £>0 ) a r a r a r ar- •i- d r 0, d0 o + ©, *>r a-, a_90 +• a0d9{ a v a r c o 5 ( to 5 t+ 90) b 3c£ Sin, ( <Out 4- = b 0 c ^ , | s^ (J L b COS ^ 4- CoScJ^t s^(j>b \ a r a r L 0 j r b. ScL 4- •!>_ Sc6.~l sCw ( u). b 4- (h.") 4- k. ^ r * c ( ,S. \- ^ A> } I a r <ar d r "£ a r 3 A5.14 + ( k e ^ l<h> + £ [" b , ^ 0 + b o ^ . l ] c o 5 ( ' o u b + ^. ) L a r L a r a r J j Ode 3) From Equations 5.62 and 5.64: a r 7 a r : a r ' (AS.?I) 3 i f 3 A5.15 ( f t • a r r / a r a-c Cud_!f> a r a r +• e \ a_a L a t r ar- ar a r oC6°> a„ bl90 4- fe f a, b^90 4- a d a^£. a r " i a r " ar 1- # 4- O^*^ ( A C ? 4 ) and Equation A5.54 can now be expanded as: aV 5 = / aV 0 av1- L a r 2 -- i " 2a_fto aj 1 a r ©1 av-1-S t ~. ( h 4 0 O ) I X 3],,, ^ e + k e 2 S ~ ( o u t 4 cf>c I a r ar a-c1-) I a r - - a r dt? V a r / a r a r a r ^ j I i[h,^o 4Sat>ae1 + a . ,d2iV 4 a o a2p, 4 9, a r • a r 1 *\, -ike. te)'I3 0 ( e l ) 3 ¥ i A5.16 As mentioned on page A5.9 , the der ivat ives of v,-(t,T) can be deduced from the corresponding resu 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 ) : d r a r dj>* = 14> (A?.??) 5v c>-c and therefore only the terms s i n 8 Q , cose Q , sincf>0 and cos(j)0 need be a l tered 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\ in •> 1 \ t r* A <~* r\ •€ P r\ 4-r\ CT -t« the so lut ion of Equations 5.78 and 5.79, which w i l l then lead to resul t s that are uniformly v a l i d when t = 0 ( e ~ ^ ) . For Equations 5.72 and 5.73 we have - O-L afc-af a r .1. — a o.^ ( A S . 71?) Jr 1-- Z- a v/ 5 + atar ar- e a ) a_v/«; a-c*-(AS . ?<)) as e 7 ( t ) = f ? ( t ) = 0 (from Equations 4.101 and 4.102), and S ? ( t ) = T y ( t ) = 0 from Equations A1.6 and A1.7 . In Section 5.4 (see Equations 5.114 to 5.117) i t was shown that a Q , b Q , eQ and <J>Q are constant, and therefore , using Equation A5.75, the terms f s V s l and [ ^ l / s T can also be 3k$ A5.17 el iminated from E* and F*. The functions j " - j and [ d_Js + c a n be assembled from Equations A5.52 and A5.53 a t a r a-c J i n the form given below. 5  <5<A^  a t a r a-c - <Hs = - u>s $ a_a. jc^Cosb+9"> +- a£i9 cos 6os b+• 9) 7 a > a r ^ L <?v 5 I a*-? a v S ( A S . 8o) • fiuf 3 t cpsfu^b +<£*) - b^>*s<~foub I a r a r r \ aV* + d** = Ja. cos (^1 + 9) - a 3 9 5i«- ( o j t < - P ) ix^ a v a r - a r -<CjoJ a«. s-*w(^t+-9*) +- cLaD^cosdo^b+e*) L at- a v ( A S . s-i) a t cesCu>tu 4-(jJ>) — t>a^ > s-^Cuijb-tcf>^) a r dxr L a r a r j As before i n Section 5.4 (on pages 5.27 to 5 . 2 9 ) , any resul 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 solut ions for the short -per iod terms. I f , therefore , 34-6 we neglect expressions containing long-period q u a n t i t i e s , Equations A5.78 and A5.79 can be wr i t t en- ' A5.18 £*(. k,"0 = Sin,od&t Z«>< , c*,$0 - n ae, <a 'O ^-o — bx a v 3 V ( A S . f f Z ) 2Q 5 i ? i ^ 9 0 4 - <itfc5£i c o ^ S c ' + cos <->s t where Equations A5.80 and A5.81 have been evaluated using the expansions given i n Section A5.2 . 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 th i s comparison f u r t h e r , we can redefine the quant i t ies Yg, Y-|» 6 n and 6, in Equations 5.94 to 5.97 by s u b s t i t u t i n g ^3' and 0Lod9, i n place of i2o and a0^° r e spec t ive ly . In t h i s case Equations 5.98 to 5.101 again define conditions for the e l iminat ion of secular terms, but from u 7 ( t ) and v^(t) instead of u^(t) and v g ( t ) . A l s o , because a Q and 0Q are constant: 2*., s^P 0 4- a.0 29, Oc 3if7 So., fco &0 - fl.0 a_£. Si^90 - 2. r ^ . ^ & o - S ^ - ^ o j a v • a r a r L J a v a r i f * , * a r L + a ^ c o D , a a , c ^ P / - a o a _ 9 ( ar L J A5.19 (AS.tffc) av ^ and we now obtain conditions analogous to those defined by Equations 5.102 to 5.105 of the form: a. ar" s - ^ f . ^ o ^ o ^ - O j V ^ o J +- £>,*[z^C^-O^-4<%^~\ . s o (ft s.ss) 5 O ( f r * . * * ? ) a. 3^8 A5.20 1 a r With the exception of and 6^ a l l quant i t i es i n Equations A5.88 to A5.91 are constant, so that we may again use the compact notation (A*.«»0' [ a? L I 3_ " •4. C 4 * . ~ C J % 8 , (AS.«S) ( c f . Section 5.4 , page 5.29), where c-j, c 2 , c 3 and c 4 are constant. Note, however, that the s p e c i f i c values of these constant terms i n Section 5.4 31+9 A5.21 d i f f e r from those for 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 solut ions are considered (see pages A5.33, A5.52 and A5.67). A f t e r s u b s t i t u t i n g for Q s from Equation A4.32, the constant terms can be w r i t t e n : c, = ZcJs s ^ 9 D | O s - - - o 3 ) + -4- a-J*- 7 (K.U) From Equations A5.92 and A5.95, and using the symmetry property to derive corresponding resul ts for b-j and <j>-j, we obtain 3a., •=. O ( A * , IOO) d_0, O (AS.IOI) 3b, •= O (AS.iox) at-av 350 A5.22 The non-homogeneous terms i n Equations A5.80 and A5.81 are therefore both zero, and consequently U.^ . Cb) = O ^ ( A s . i o O 4. \Jf Ct) = O ' . . t>s. ie>S) We now proceed with the s o l u t i o n of Equations 5.74 and 5.75 f o r Ug and Vg. 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^" t l i + _ ! l £ i i ! _ 7 sc«.t - j 5 s£«-3b *" [ I 16 C H - p,> J 32- J 1 cos t + IS cos 3 b ^ 2 S i - v 3 b ^ + 3 - tf cos 3 b ^ l CAS ."ot.) t, Note that u . ( t ) = u - ( t ) . l u 1 I \\W v . ( t ) = V i ( t ) N H for a l l i > 5 (see the discussion on pages 5.16 and 5.17 ) 351 A5.23 tosb - Z$ cosZb 64-(AS. lo?) At present (from Equations 5.114 to 5.117 and A5.100 to A5.103) 8 a = 0(e 2), f £ = 0 ( e 2 ) , | | = 0(e2) and | ^ = 0 (e 2 ) . Consequently 3T neither dVs or aVj- contribute at 0(e) to Equations 5.74 and 5.75 dxx a t * (see Equation A5.54). The functions - 2 | ^ s - }Js | and \_dbdv bv J oce z) -2 f + ^ | r x , which can be evaluated from Equations A5.80 and L btbr jOLeJ _. and A5.81, take the fo l lowing form: - 2 ( A S . l o g ) Son. (O st a r 3 i a v a-r I m; a r 3 t a t l <*>S>0 - Q., air aj>z a-r j I atr aV J <-av - t o * * * 352 a t a r a r A5.24 + co5 «0st + Cos C0 ut e J 1 ar j a r a r L a r a r I <?r a r J L 3 IT a v z ^ ^ J att 5 ^ ^ * + b , ^ c o s ^ o * 2 - z f ab,. «*>fo ~ b o ^ ( a r a r J L a r a r J Note that i f 1 * . and a 0 SO. in Equations A5.82 and A5.83 are ar a*c replaced by and a 0 Z9i. r e s p e c t i v e l y , the r e s u l t i n g expressions at* <>T are i d e n t i c a l to the corresponding short-period components of Equations A5.108 and A5.109 above. We shal 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 5.75, a*g c - av s at- bt 0(1) and 20+- p , ) 1 a Vg + au.^ at 7- d t J o ( i ) include both homogeneous and non-homogeneous solut ions for u 5 ( t , x ) and v 5 ( t , x ) so that U j C b . r ) « a-tt&Cus t + + • b Si*-Cu>Lk+0") -+- U 5 ( t ) N r t ( A S . U o ) 353 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 5.85): A5.25 ( A S . 112.) ^ 1+ p, ) i C ^ 5 l - ^ X ^ ^ : L ( ^ - u ^ x v - o ( A 5 . " 3 ) «O u.5 20^s^cO ub + <->,_0,.c<ocO,_b ? + / 2 o 5 sc^ .b + O x Cob 351+ A S . 26 ^ 3 H« " r ~i ( A S . U 4 ) (^-•vx^-o " 4- J ZOj.tj^ b -^3+o()sCv k j -5 f . . N [ r . A , r "\ 1 . . .s V 1 S (1+ p,)^ E C s ^ - ^ X « V - 0 ( A S . " 5 > These equations lead , a f ter some rearrangement, to the fo l lowing expressions: 355 A5...27 { Cos u>s b 4-^ . f -/>. . . . . . . 1 . y 2.1 ~~? 7 J t 7 - J b - £ 356 A5.28 4-I) Note that the usual symmetry between short and long-period terms i s preserved here ( c f . page 5.23). For the homogeneous components of Ug(t ,x) ana Vg ( t ,x) we have: St*- Ti ««?sv a . Cso(jo>s b + 9) - <CS w s a . sc* _ ( c o 5 b •+• 0 * 0 + bcoCioub+-c^>) - <QU^U bs^*- Cto u b + <£>*) *b x Xt ( AS. i l<?) + UJUD **•«.(* «oub +j>} + fflL«*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 into the analysis for higher orders of e (see Equations 5.76 to 5.79 i n Chapter 5) . I t 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 error 0(e ) , which i s the accuracy required by Equations 5.75 and 5.79. The process of expansion i s s t ra ight forward, and from Equations 5.62, 5.63, A5.57 and A5.58 we obtain the expressions given below. < r — <^.S ( A S . i x o ) - "J'-bo j ^ Outc^j>o - C u , b v L ^ 0 ^ . +• ^ u ° u b 0 ^ s^«j ub +• «*> o t b * c ^ 0 * ^ b, £ t w ^ t c o ^ - sc~«Jufc s^ ^ c . ^ + ^ J " b0<^>( ^ w o ^ b s o ^ +• co^^b * ^ 0 ^ 358 A5.30 z - [ a - - - a " ^ J c o s ^ t + e,,) + ^ J Q . e . - t - a c e i J j ^ c ^ u e j + Q s s | f t x - o - c f ^ j $^Co 3t+• + Q s t o s j a , © , " T " A ^ 9 ^ ^ C ^ k + e * ) - t 0 ^ j a > s C t J u t + ^  + t>0^ j s^(o^b4.^) L ^ a t J H . (f>f. H I ) fcsc-9: - o 5 a , ^ v i ^ f c c « P e + - < i o S b C P 0 ^ - ^ s a 0 © , ^ 359 A5.31 Note again the symmetry between short and long-period terms i n Equations A5.120 and A5.121. This symmetry i s exhibi ted by a l l non-homogeneous expressions in Equations 5.74 and 5.75 that involve u) s and 0^, and we may therefore consider only short -per iod terms when accumula-t i n g 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 also be ignored, as the functions { S 8 c - 0 - 2 k , ^ ^ ^ g c t ) j and I T ? a ) - Z k - , ^ e g C t ) J generate no secular terms i n the solut ions for Ug ( t ) N H and ^ ( t ^ H ' W ^ t h these s i m p l i f i c a t i o n s we can w r i t e : 3 6 0 A5.32 I a r at? j (_av a v + 2 e ' s C n - j O H i - O + Ceo l <)v av .5 c at? atr i - - - V (^/-"VO ~~~ (fl5. 127.) f 1 - ^ ^ ! ^ + -2.6 + ^,)"«-1 cOg 6^* - cos a.0 fe P 0 ? + C 0 5 t O $ t 361 A5.33 ] ox bV J> \ bx ar J •15 k,^ / * , ^ , 0 - / * / { ^ [ 4 4 - o , - o s ^ f - 2 o 3 7 -ICU^HQ,^ I f secular terms are to be el iminated from Ug(t) and Vg(t) Equations 5.98 to 5.101 must be s a t i s f i e d , where YQ, Y-| » <5Q and 6-j denote expressions i n Equations A5.122 and A5.123 when we take A f t e r some rearrangement, Equations 5.98 to 5.101 lead to the fo l lowing condi t ions : - i e ^ O + ^ . ^ C i - k e ) | ^ ( ^ - O + 0 ^ 362 A5.34 (ft*. av -if W,/^  ^ j / ^ C i - ^ o " 1 V [ 1 fc»50^«toOo - ( c J j 1 - * . 0 , ) s ^ P 0 363 A5.35 where c-j, and are defined on pages A5.20 and A5.21. In der iv ing these expressions, some reduction has resulted, from the e q u a l i t i e s (AS.1*1) which correspond to Equations A4.7 and A4.8 when a ) ' and io L s a t i s f y Equations A4.28 and A4.29. A f t e r subs t i tu t ing for 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 wri t ten as: 2 u 5 ( <v - ar 4. 0 *i«-80 39* ? ( A * . i 3 c O ( f t S . l * l ) CAS. I32- ) 36ii a v j A5.36 where the constant expressions d . ( i = 1,2,•••4) can be arranged i n the form given below: «o. /L. *- ' ' ( A S . 134) ( A S . J 3 4 0 365 A5.37 -In der iv ing these values from Equations A5.124 to A5.127, a substantial reduction has resulted from using Equations A4.32, 5.106 and 5.107 for Q s , sin0Q* and cos9g* respec t ive ly . Now because Equations A5.130 to A5.133 involve only two un~ u e t t i r ' n i i l i e u q u a n o i u i e b , o^x. a u u " ^ t , uwu u i u i e s c i u t t i c<-|uu o i bX ZX should be redundant. We can rewrite Equations A5.132 and A5.133 as: i ., • i = 3L. (flS.I3S) .f. Ctn>96 (AS. 131) 366 A5.38 and therefore: ( A S . K l ) The expressions C ^ i 3 - 2 o s i v ) and (2«J»si3 + v ^ ) can be arranged i n the form: -4e~56+j0*(l.-0 C^- 0,) fAS.lA-3) A l s o , from Equations A5.128 and A5.129: 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 write Equations A5.130 and A5.131 as: 368 A5.40 then from Equations'A5.147 and A5.148 i t fol lows that - i 5 k 6 S. ISO With the exception (at present) of ^ and a 0 3 P Z , a l l terms i n a*r at these expressions are constant, so that 369 A5.41 ^•2. - constant , ( f l ? . i f i ) and a l s o , i f a 2 (x) i s to s a t i s f y the condit ion of uniform v a l i d i t y i n T: daz. = O. (As. i s i ) S i m i l a r l y , using the property of symmetry, we can wri te = constant OKi?3> bx oxr Eauation A5.152 Drovides. toaether with Eauation A5.149, the necessary condit ion to determine 8Q. In p a r t i c u l a r , i f ( l -kg) = 0, we obtain the fo l lowing r e s u l t : and .by symmetry: Corresponding values for 9Q and <J>0 can obviously be obtained when k Q } 1, but not of such an elegant form. At t h i s point i t i s tempting to assume that 2[£t = 0 (see Equation dx A5.248), and from Equation A5.150 we then have 370 A5.42 The short -per iod components of u 5 ( t , x ) H and v 5 ( t , - r ) H a re , 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 < IT. A f t e r expanding cos(w s t + 6Q) and Q s cos(w s t + 0 Q * ) , from Equations A4.32, 5.106 and 5.107 we can w r i t e . (ASM 6°) <C$ftoCcosb 4- ^J^CeoPo - Z O j S ^ P o ^ = Co o? 3 k r Z ( 0^O% 4- cOx 0 X ) 9 0 90 4 - 4 - S j J " &0 - S^. t J sk ( « S l - 0 3 ) L (As, 371 Now for e i ther value of 9g: r P = 'if 0* ( A S . l t * ) COS i n which case A5.43 cosCw 5 t + 6 > 0 ) { < ° A c o 5 9 o - 2 0 3 s in . e e j . = t ^ O ^ t o O j t ^ Z o 5 $ ^ t - (AS. IAS) -•^s C o , - ^ * ] ceocojfc - 2-O t sc^to 5 t . (A*, ut) I f we set kg = 1, then from Equations 5.84, 5.85, A5.157, A5.165 and A5.166 the complete solut ions f o r u^(t) and v^(t) can be reduced to the fo l lowing form: U. $Cb)= - l $ l f , ^ u 5 20 3 K v ^ t + Oncost £ ^ , | * x ( l - / A » . ) X +• (AS . I t ,^ ) I ( ^ -1X0^-1 V -) sO+fO"*\ v $ Ct) = lSklL?;L0»- S l w t r * (-Z + Oj) cost 7 ^,^xCl-/jQa + OU) ( A M f c g ) 372 A5.44 where the symmetry property has been invoked to el iminate long-period 5 terms. Note that , 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 solut ions for higher orders of e. The present analysis i s s u f f i c i e n t to determine eu(t) and ev(t) o in Equations 5.32 and 5.33 to wi th in a constant error 0(e ) , although the solut ions are uniformly v a l i d f o r t = 0(e"^) as e ->- 0. There i s conse-quently nothing to be gained by evaluating u i ( t ) ^ H and v . ( t ) N H . f o r i > 7. Before we continue with the s o l u t i o n of Equations 5.76 and 5.77 for Ug and Vg, the expansions of der iva t ive terms must be extended to include higher orders i n e. At t h i s stage of the a n a l y s i s : da. = £ 3 ^ 3 +- e^}a.H 4- OCeS) (hS. Ifcq) b~F bV b& + bX~ ?9 bx1- Zx*-and therefore from Section 5 .2 : 3a. coiC usbi-9) 0X bv ( A S . 1 7 3 ) ««.(«,t+9 0) 4- B 3a 3 Gos(eJjt+ 90)~l +OUs) Ib-x 373 A5.45 i X a. 0 d & i c«o £ <->, b -t- S o ' ) av + £<„ ? Ceo (o} L a*c ar j atr I dv av a x j L dv av j (AS. 174) av + .e-L ar a ^ j av I a-r ar ^r) 6 at- at- 3 a r x. ar-CAS .17S) a 0.5 a^ ,• - e a\-j c o ( i o Jk+© 0 >) - d 0 ajj&j si«.r"u>,b+ © e) ax1- at"1-a^ a0a2fs * M . ( * « J , f c * 0 , at*-CAS.I7?) 37k A5.46 Note that , i n the in teres t of b r e v i t y , long-period terms have been omitted from Equations A5.172 to A5.177, as complete expansions inc luding the long-period components can r e a d i l y be obtained by inspection from these resul ts (see Section A5 .2 ) . The functions 3_j*s and \js consequently do not contribute a v z to1 o to Equations 5.76 and 5.77 at 0(e ), and for the non-homogeneous terms we now obta in : CAS For the remainder of t h i s analys is s p e c i f i c non-homogeneous solut ions are not required (see the discussion on page A5.44). I f secular terms are to be el iminated from Ug and Vg, we may therefore r e s t r i c t our at tent ion to quant i t ies i n E* and F* that involve short -period terms alone, since corresponding long-period resul 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 u 5 ( t ) ^ H NH 5.33). and V g ( t ) N H can be w r i t t e n i n the f o l l o w i n g form ( c f . pages 5.31 to 375 A5.47 lL , ( t ) = - JA x Sin- 6*)s t - 0 . ( A S . 1 8 0 ) r o s(o/-- u j-y - | t , <vo 4 j s b J O f i ^ - ^ k . ^ C - z O r 1 1 - 2 . ( ^ - 0 ^ 1 - ^ [ i f ^ - o , ] and i f Equations A5.180 and A5.181 are re-s tated using the compact notation ( « V - - < o j O 376 A5.48 then we can wr i te : if- *b ( R S . I S 4 ) 5V t + 3 a b » s S ^ 0 5 b C ^ - > 5 ' ) H- c « O s b - X, ) . (fls.iss) From Equations A5.52 and A5.53, together with Equations A5.172 to A5.175, and again considering only short -per iod terms: L CAS. 1 8 b ) 5.^ 0 s b 1os <to&. 15 377 A5.49 - Z. (A?. 187) a v - S^0o i f >*cr \ 4 C<o tOjk z<&5 o 51 - z. J «te er r We are now i n a pos i t ion to evaluate Equations A5.178 and A5.179, as the remaining quant i t ies 2 - | f V 5 - J y , — ^vj c and have already been determined (see Equations A5.120 and A5.121). A f t e r some rearrangement E*( t ,x) and F* ( t , x ) can be wr i t ten i n the form given below. 378 A5.50 a r a r . 1 1 L >r >r\ i / zci+^y*- < D } < j i<i e E , i + *<ps 2a, - * 0 M A L L a r a - c j + <*>P0 L a r J j I L ^ r J 379 A5.51 Sin. u>,b + 1 QQ-i — a. bf + z ^ O t n . y 1 s £ x^ - x3*> +- COS u>st ar ar -«r>9. (_ I bV dV a r a r J J 380 A5.52 I f these equations are wri t ten as ( A ? . I<»l > then we can again use the conditions given i n Equations 5.98 to 5.101 to e l iminate secular terms from Ug and Vg. A f t e r some rearrangement the fo l lowing resul t s are obtained: ^ 3 - ^ R ^ av J V iv . iV ^ 3 -*o°*>j>z IX bv - c, at av CO 381 A5.53 , r + 4 ^ 3 - a 0 D , ar av 0>* 1 - ) a P. ar - c 3 Q |a&x 4-i ar -3 8 2 A5.54 - 2 U S ( I + J I , ) * 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 wri te l at- 5 I atr 3] (ft 5.1*7) ( ^ , - a_pt 7 r ?^pe 4- z ^ s 0 0 1 - | *x,3£ t +ftoa^ ] [ Z « a , < . P e - 0 t c o P 0 ] I av 3 L at- a v j S£^ .90 + X ^ Ceo 0 o 1 L a r ar S (flS'.m). 3 8 3 A5.55 where the quant i t ies ( 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 resul ts of Equations A5.145 and A5.146, to demonstrate th 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 deta i led c a l c u l a t i o n i s omitted here, both of these condit ions are , i n f a c t , s a t i s f i e d . Before so lv ing Equations A5.196 and A5.197 f o r a 1 and 6 - | , i t would be helpful to reduce the functions H 1 and H 2 to a more compact form. A f t e r s u b s t i t u t i n g for Q s , s i n 8Q* and cos 9 Q * (see Equations A4.32, 5.106 and 5.107 respect ively) . the c o e f f i c i e n t s of aQ8^ and a^ i n Equation A5.192 can be expressed as: (RS.20Z) 381; and for Equations A5.196 and A5.197 we now have: A5.56 ( A S . 203) 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 ] ( f t S . Z o S ) 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 process of reduct ion. 385 A5.57 The expressions involv ing ( i = 1,2,•••4) may be re-grouped, and Equations A5.204 and A5.205 then lead to the fo l lowing condi t ions : ^ 3 bX dX a r u e ^ x . E^ - O j - z ] (05.26?) Now from Equation A5.151 _^S>z = constant, so that both da.3 a t and 39 3 must be constant. In th i s case we can w r i t e : a r L ^ i r" w 5 c to^ 1-<JJ"!) L (fl?.20S) 3 8 6 A5.58 and, i f a~(-r) i s to s a t i s f y the condit ion of uniform v a l i d i t y i n T: Sen. <9e £ ^ + X ^ O j 1 [ W s l - 0 3 - Z 1 ( A S : io<0 Without d e f i n i t e values for 3_ea and ao 3 Equations A5.208 ot at and A5.209 cannot be solved for a-j and B - j , but i f we assume that both 02 and 6^ are constant: ' t f 5 JivPo^ [ ^ - O j - Z ] + ..... ^ CflS.Zio) 0 ,<VoPo^ X, [ ^~Oi - 2.] - • ^ 387 A5.59 where a Q i s defined by Equation A5.157. Note that , corresponding to the two possible values of 0Q wi th in the in terva l -IT < 6Q ^  IT , there are two values of 6^. The functions involv ing 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 re ta in 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 re la te only to short -per iod terms. Di f ferent values of X^ consequently apply i n the long-period case, and t h i s must be taken into consideration when evaluating Equations A5.208 and A5.209 with b , <f> 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 for U-|Q and V.|Q. Only those quant i t ies which generate secular terms i n the so lut ion need be considered, so that the functions . A D < t > - 2 k , | ^ $ l o a > and T l D C b ) - Z k . ^ c . / t ) ' may both be ignored (see Equations A l . 6 , A1 .7 , 4.101 and 4.102). From Equations A5.100 and A5.151 both a, and d j i are constant. Consequently, from Equations A5.197 and A5.198, 3f-3 and a_93 are constant. In t h i s case Equations A5.176 and A5.177 reduce to the fo l lowing form: av 1-This s i m p l i f i c a t i o n i s discussed on page A5.46. 388 A5.60 and therefore neither of these terms contribute at 0(e ) to Equations 5.78 and 5.79." At th i s stage of the analysis a ~ and | | are o(e2) and 0(e ) respect ive ly (see Equations A5.151 and A5.152), and from page A5.22 Uy(t) and Vy(t) are both zero. Consequently the der ivat ives of u^ and v 7 , together with the expressions contr ibut ion 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,^ then from Equations A5.52, A5.53.and A5.172 to A5.175: . bxr A t c f . page A5.46. 389 A5.61 2o< I ] I • z.7 a jv s + a^. 7 I a t - a r a r 3 CflS.2.1S") 2<3?, £05 ^ C o 9 { dr a r LP* "1 a r \ - s ^ D 0 * [ a i . a 3 + + «. e f aj^. - & * a & A +^3*3! L a r a r v ^  d i r > ; ^ J 3 9 0 A5.62 Note the equivalence i n s tructure between these two expressions and the corresponding resul ts for lower orders of e (see Equations A5.82, A5.83, A5.108, A5.109, A5.186 and A5.187). . We next require expansions of : t and at t h i s point the advantages of assuming k n = 1 and 0 are con-u bX s iderab le . 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 fol lows that none of these quant i t i es 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 ex tens ive ly , as shown below. - z d <XS a v i atx at S oaO (PL*.211,) Note that u 5 and v g here denote complete s o l u t i o n s , so that u 5 = u 5 ( t ) j | + u 5 ( t ) N H v 5 = v 5 ( t ) H + v 5 ( t ) N H ' c f . page . A5.24. 391 b\s + du.s a t a r }tr fOL-f a \ / 5 -t- Ji* A5.63 ^ W e ' ) 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 obta in : - z |_ yc a r V a r 3- a r / a r J 3 L *v ar ar J j - Z arc a r i t av ar (AS". 218) 3 9 2 A5.64 r av <?ir a v j 3 + z fe- ] 1 - x h $ * } } ] A f t e r su bs t i t u t in g for Q G, s i n 6Q* and cos 8Q* from Equations A4.32, 5.106 and 5.107: 393 A5.65 © 5 u>5CJO +5^9,, = u>^0xcos9b - s;^90 ( u ) 5 ' l + o 3 ^ , CAf.^2-3) so that Equations A5.218 and A5.219 can be wri t ten as: 1 39i+ A5.66 F * ( t , r ) 4 co^jb! I < P 0 -P e + <te0 < >C t V+ o 3) ? (AS. 12-5) where H L i (fl*.2Zt) L ar y a^ - e, 1< ^ r * ^ 3 ] a r j (ft&zz?) L M « 0 e , i?3 - C*0e>. +<*tO ar ae* ] ar J I f we now wri te E*( t ,x ) and F * ( t , x ) i n the form 395 A5.67 F * ( t , t ) = o~b Sir*. £0j t -v J ( COS CO$ t , C AS-.zs.i') then to el iminate 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 . Af ter some straightforward manipulation the fo l lowing t conditions are obtained: CflS.2.30) o - o (AS".i32.> t ^ 5 C* tOf - t^t?} (AS.233 ) By comparing these resul ts 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 respec t ive ly . 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 wr i t ten as: c, Ux - c,, 8, = o c z + t , H, = 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 s H i . ~ c 4 H , = ° ( AS.2.36) which i s consistent with previous resu l t s (see page A5.20) . From Equations A5.230 and A5.231 we now have: H , C O = O C A S - . z S g ) H x £ r ) = O , (AS . i 3 < 0 so that ..... <?r . . . a_tr a r . _ _ w j e 2- a r ^ [ 3. J . (AS.Z4.0) a r a r L J (AS . 2 M ) At the present stage of the a n a l y s i s : ar £ § 3 = constant ar (from page A5.57) , and from Equation A5.151 Z9Z = constant 397 A5.69 From Equations A5.240 and A5.241 i t therefore fol lows that ^f-q. = constant (AS.a-4-i-) bO^ = constant . 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 wr i t ten as: a v bv a r We can subst i tute for a Q i n Equation A5.246 from Equation A5.208 to obtain: CAS. 24?) Ceo - « , s ~ . P < > ^ \ [ t O ^ - O j - 2 - ] + . - ^ 398 A5.70 and therefore, since the function j^Ci + p,) *• -.|P». ^ i ^ - "I is constant and non-zero: 0 , - constant ( « i * * < 0 (see Equation A5.209). I t i s convenient at t h i s point to set and from Equation A5.247 we now have: P ^ - /*,P, ( T c ^ P 0 - tf^s^Q ... . ( f t r . ^ i ) . . • a . « 5 V - C O J ) " I f i t i s assumed that both ^ 6 3 and 19^ are zero, then from at at Equation A5.245: 2. where, i n th i s case, 9-j i s defined by Equation A5.211. Corresponding resul t s for b 2 and (J>2 can r e a d i l y be obtained using the symmetry property, but note that the quant i t ies 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 variables o^, b and-<j> (see the discussion on page A5.59) . 399 A5.71 This completes the deta i led analys is of four-body motion near and Lg , although the resul t s derived here are condensed i n Chapter 5. A6..1. APPENDIX VI CHEBYSHEV POLYNOMIALS A6.1 Introduction Suppose we wish to approximate a funct ion f (x ) over some i n t e r v a l , for example - l ^ x ^ l , to wi thin an error ±e using a polynomial i n t e r p o l a t i o n . 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 i s to be maintained over the ent i re range of x . In t h i s case an i n t e r -polat ion procedure based on Chebyshev polynomials leads to the most "economical" approximation, in the sense that , for a cer ta in accuracy The Chebyshev polynomials T k (x ) are simple trigonometric functions cosk6 s but expressed in terms of the var iable IX- = cosB , where -IT ^  6 ^  IT and, consequently, -1 ^ x ^ - 1 . I t i s frequently * more convenient to work with the sh i f ted Chebyshev polynomials T^(x) , for which T * C x ) = cos k9 k with cos 9 =- 1 2+01 A6. As 6 varies from 0 to i r , we now have 0 ^ x ^ l . The method of approxi -mation for an a r b i t r a r y funct ion f (x ) i s out l ined in the fo l lowing sec t ion . A6.2 Function Approximation using Shi f ted Chebyshev Polynomials Consider the problem of approximating some funct ion 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 var iable y must be defined such that the in terva l -a £ x ^ 3 corresponds to 0 ^ y 4 1. Of the many t rans-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 in to an even funct ion of 6 which can be expanded as CO . . . rK*0 = + j . % cos\< & . (06.3) l e s t --However, since T|<(,y) = coske, t h i s expression i s equivalent to 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 [621*]. and i n pract ice the summation may be terminated a f t e r n terms. I f |am| denotes the magnitude of the larges t c o e f f i c i e n t , the summation could be truncated when, f o r example, |an+-|l < 10 1 0 |aml • We then have Z — i k s i where e i s s u f f i c i e n t l y small that i t s e f fec 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 t j ^ = 1 + cos C S " T T/V) S = Of \ > 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 of data points 6 over the i n t e r v a l 0 6 is- TT. Equation A6.5 can be rearranged into the power ser ies n. . . .. . iAx} = N ' t \ . . . . + e. , ( A 4 - 6 ) 21—» and the corresponding series i n x can now be w r i t t en ^ o r , a l t e r n a t i v e l y : J where k03 denotes the binomial c o e f f i c i e n t m A6.4 For computation, however, the fo l lowing form of Equation A6.8 i s more convenient: J A6 3 Coef f i c ients of (y) I f the sh i f t ed Chebyshev polynomial of order k i s wr i t ten as k i — O 2-. then, from Lanczos [ 6 2 ] , the c o e f f i c i e n t c^ i s defined by: W.-C -( A * - II ) Coef f i c i en ts of the f i r s t twelve s h i f t e d Chebyshev polynomials are given i n Lanczos [62*] . 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 A6.5 the error e may be taken as zero. 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 A6.6 , we can also wr i te as k £6c"> - t k <j . ( f i t . 13 > k=.o Now from Equation A6.10: k so that k H k * n where |T^(y)| ^ 1. I f we subst i tute for y i n Equation A6.13: which can now be wri t ten i n the form A6.6 CA6.I7 ) 1 5 1 • - . C A 6 - I 8 ) This process of contract ion can be continued so that f ( x ) i s approximated by progressively fewer terms, but with an increasing error [62 7 ] -We next invest igate the behaviour of |S| when n-1 terms are retained but the in terva l -a ^ x ^ 3 i s decreased. I n t u i t i v e l y one would expect the error to d i m i n i s h , and i n fac t the reduction i s associated with b . n Let DC = = c 0 •+ CT£ (M,. |q ) where a t l , the in terva l of i n t e r e s t i s defined by -oo. ^ 5 and f (x ) i s bounded on t h i s i n t e r v a l . As a.-»• 0, f ( x ) w i l l tend to some value f ( X g ) . We can therefore wr i te 4Gt> = S C O + v$*C$) , Cfl6.ao) it • where f (£) i s bounded and f (x) - f ( x n ) = 0(a) as a -*• 0. The in terva l 1+05 where the error 6 s a t i s f i e s A6.7 - a ^ C - B corresponds to and subs t i t u t in g for x i n Equation A6.1 we obtain ^ - % + oc ) The function cj>(6) i s now defined by which we can wri te as Equation A6.13 i s equivalent to rt. (see Equation A 6 . 5 ) , where the Fourier c o e f f i c i e n t s a k are derived from a numerical i n t e r p o l a t i o n process. Note that , because n i s the larges t exponent of y i n Equation A6.13, b n i s equal to a n < If , instead of 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 a^ were obtained from the in tegra l IT • « k = i . | (f>C0) coSlt& d9 , ( f l 6 . a - 5 ; o . 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, in which case Equation A6.20 would be exact [62 8 ] - Although t h i s condit ion may not be s a t i s f i e d , we assume here that Equation A6.25 can be used to define a n and b n > Consequently we can wr i te If (9) cosr\9 <19 . , . and therefore , from Equation A6.22 H II Cos T? J i r For b we then obtain n i r Cos m. 9 <&9 (fl&-2s) so that b n = 0(a) as a 0 . When n » 1, the quanti ty c n i n Equation A6.18 w i l l be a large integer ,^ and i t fo l lows that \l\ = O ( r ) as 0- _> o The error 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 XQ - oa ^ x XQ + a3 therefore diminishes as a -> 0. From Equation A 6 . l l : c n c 4 0 8 A6. Note that , for 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, i f g(x) denotes the approximating polynomial : I f ( x ) - g(x) | ^ |5| for a l l x such that x Q - aa ± x ^ X Q + a3, so that f (x ) i s approximated uniformly over the ent i re in te rva 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 about i s given by: J _ r 1 + and i f we take ... -x.. . = then . . [ 1 + For L . and Lj- | r . . | *= 1 (Equation 3.45) , and i f f ( x ) = _L we have k-09 A6.10 = [ I + J ( A 6 . 3 l ) 2 where x = | r^. \ - 1. I f f (x ) i s approximated i n the in terva 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 wi th in a region about p . , bounded by inner and outer r a d i i ( l - a ) 1 ^ and (1-3) f x r espec t ive ly . For the analys is 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 t e r v a l s i n the sense that a and 3 must be equal f o r both expansions. 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 ( i = 0, 1, 2, 3, •••• ) i n Equations A5.15 and A5.16 would d i f f e r . The corresponding region of the r\ plane near p 0 and p~, wi thin which the expansions of _ j and , ,3 are both v a l i d , i s shown i n Figure A6-2 for ( l - a ) ' 7 * = 0.65 and (l+B)' / a = 1.35. 14-11 A6.12 Note that i f a and 3 are chosen so that l - ( l - a ) ' / a " = (1+3) / z -1 = p, the expansions are v a l i d ins ide a c i r c l e of radius p about and Lg. For the function f(x') = [1+x] , 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 error e in Equations A'6.5 to A"6.9 i s reduced below T 0 ~ ^ for 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" over the in terva l -0.5775 < x 4 0.5985. A s l i g h t error becomes apparent for 0.5985 ^ x ^ 0.8225, reaching a maximum of 4 x 10" 7 when x = 0.8225. The c o e f f i c i e n t s of xm in the expansion of f (x ) are tabulated below f o r fl y tvi S I E 4- A n ^  +- U ,.,^4-U o^wv*^/.^««rl4^r. U 4 « -I 1 * . 1 . . ~ <-A6.13 Chebyshev - Derived Binomial C o e f f i c i e n t s C o e f f i c i e n t s 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 Coef f i c ients of x m i n the Series Expansion of [1+x] for -0.5775 $ x ^ 0.8225 PUBLICATIONS Barkham ; P.P.D. and Soudack, A.C., An extension to. the method of 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., Approximate s o l u t i o n s of n o n - l i n e a r , non-autonomous second-order d i f f e r e n t i a l equations. I n t . J o u r n a l of C o n t r o l , V o l . 11 (1970), pp. 101-114. Soudack, A.C.', and Barkham, P.G.D. , Further r e s u l t s on "Approximate s o l u t i o n s of no n - l i n e a r ^ non-autonomous second-order d i f f e r e n t i a l . equations." I n t . J o u r n a l of C o n t r o l , V o l . 12 (1970), pp. 763-767. Soudack, A . C, and Barkham, P.G.D., On the t r a n s i e n t s o l u t i o n of the unforced D u f f i n g equation w i t h l a r g e damping. I n t . J o u r n a l of C o n t r o l , V o l . 13 (1971), pp. 767-769. Barkham, P.G.D. and Soudack, A.C, Approximate 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 . n o n l i n e a r i t i e s . I n t . J o u r n a l of C o n t r o l , 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 to four-body dynamics. Presented at the 13th I n t e r n a t i o n a l Congress of T h e o r e t i c a l and A p p l i e d Mathematics, Moscow, August 21-26, 1972. 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 to four-body dynamics. Presented at the 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 Congress, Vienna, October 8-15, 1972. 

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