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The motion of a self-excited rigid body Lee, Richard Way Mah 1964

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THE MOTION OF A SELF-EXCITED RIGID BODY by RICHARD WAY MAH LEE M . A . S c , Univers i ty of B r i t i s h Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Matheraat i cs We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1964 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and stu d y . I f u r t h e r agree t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be al l o w e d w i t h o u t my w r i t t e n permission® Department of Mathematics The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 , Canada The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of RICHARD LEE B.Sc, The University of Manitoba, 1956 M.A.Sc., The University of B r i t i s h Columbia, 1958 MONDAY, MAY 4, 1964, at 10:00 A.M. IN ROOM 125, ARTS BUILDING COMMITTEE IN CHARGE Chairman: F.H. Soward External Examiner; J.B. Diaz I n s t i t u t e for F l u i d Dynamics and Applied Mathematics, University of Maryland A.H. Cayford E. Leimanis E. Macskasy D.C. Murdoch G. Parkinson J.F. Scott-Thomas THE MOTION OF A SELF-EXCITED RIGID BODY ABSTRACT This thesis discusses the motion of a r i g i d body about a f i x e d point subject to a torque a r i s i n g from i n t e r n a l r e a c t i o n . Such a body i s c a l l e d s e l f " e x c i t e d , and i t s motion i s governed by E u l e r ? s dynamical equa-tions o F i r s t , i n Section 2 we consider the case of a torque vector which i s fixed in. d i r e c t i o n along the largest or smallest p r i n c i p a l axis of the body s and has a component in. the chosen axis equal, to a constant plus a perturbation, term that i s proportional to ;the square of the modulus of the spin vector GO(p,q 3r)„ I t i s shown that Euler's equations can be integrated i n terms of a v a r i a b l e <f> , introduced by means of a d i f f e r e n t i a l r e l a t i o n . Further quadrature and inver-sion y i e l d p } q s and r as functions of the time to Using the method of phase-piane analysis, we show that the spin vector can perform a v a r i e t y of -motions with respect to the body-fixed trihedral° In p a r t i c u -l a r , when the perturbation i s zero, we i n f e r from the corresponding phase-plane t r a j e c t o r i e s that the spin vector can perform asymptotic motions of the f i r s t and second kinds and p e r i o d i c motions about permanent axes l y i n g i n the p r i n c i p a l plane perpendicular to the torque vector. Some of the r e s u l t s for t h i s case were also obtained by Grammel, using d i f f e r e n t method. In the general case, when the perturbation i s not zero, these motions are preserved. However, a second type of p e r i o d i c motion e x i s t s ; i t occurs about the p r i n -c i p a l axis containing the torque vector, the p r i n c i p a l axis i t s e l f being a d i r e c t i o n of stable permanent r o t a t i o n . In Section 3 we consider the same problem with the torque vector acting along the middle p r i n c i p a l a x i s . Using the methods of the previous Section, we show that o5 can assume pe r i o d i c motions as well as asym-p t o t i c motions of various kinds. The p e r i o d i c motions established i n these two Sections are then computed i n Section 4 as power series i n a small parameter. F i n a l l y 5 i n Section ,5 the motion of a symmetric r i g i d body moving i n a viscous medium subject to a time-dependent torque i s studied„ Its motion i s com-pared with that i n a vacuum. We show f i r s t that p s q , and r can be expressed in, terms of c e r t a i n i n t e g r a l s . For the special case where the self-excitement i s time-independent and f i x e d i n direction within the body, these i n t e g r a l s can be reduced to the generalized sine and cosine i n t e g r a l s , Their values can be computed from asymptotic and power series which are developed i n the same Section. The asymptotic behavior of the spin vector" i s then discussed;, y i e l d i n g q u a l i t a t i v e r e s u l t s which are summarized i n three theorems. F i e l d of Study; Mathematics Nonlinear D i f f e r e n t i a l Equations I E. Leimanis Theory of Functions of a Real GRADUATE STUDIES Variable D. Derry Nonlinear D i f f e r e n t i a l Equations I I Topology Theory of Functions J.F. Scott-Thomas S. Kobayashi F. Brauer C e l e s t i a l Mechanics E. Leimanis Related Studies: Unsteady Flow Hydraulics E. Ruus E l a s t i c i t y J.A. Jacobs Theoretical Mechanics J.C. Savage i i ABSTRACT T h i s t h e s i s d i s c u s s e s t h e m o t i o n o f a r i g i d body a b o u t a f i x e d p o i n t s u b j e c t t o a t o r q u e a r i s i n g f r o m i n t e r n a l r e a c t i o n . S u c h a body i s c a l l e d s e l f - e x c i t e d , and i t s m o t i o n i s g o v e r n e d by E u l e r * s d y n a m i c a l e q u a t i o n s . F i r s t , i n S e c t i o n 2 we c o n s i d e r t h e c a s e o f a t o r q u e v e c t o r w h i c h i s f i x e d i n d i r e c t i o n a l o n g t h e l a r g e s t o r s m a l l e s t p r i n c i -p a l a x i s o f t h e b o d y , and h a s a component i n t h e c h o s e n a x i s e q u a l t o a c o n s t a n t p l u s a p e r t u r b a t i o n t e r m t h a t i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e m o d u l u s o f t h e s p i n v e c t o r o8(p, q, r ) . I t i s shown t h a t E u l e r ' s e q u a t i o n s can be i n t e g r a t e d i n t e r m s o f a v a r i a b l e tp$ i n t r o d u c e d by means o f a d i f f e r e n t i a l r e l a t i o n . F u r -t h e r q u a d r a t u r e and i n v e r s i o n y i e l d p, q, and r as f u n c t i o n s o f t h e t i m e t . U s i n g t h e m e t h o d o f p h a s e - p l a n e a n a l y s i s , we show t h a t t h e s p i n v e c t o r can p e r f o r m a v a r i e t y o f m o t i o n s w i t h r e s -p e c t t o t h e b o d y - f i x e d t r i h e d r a l . I n p a r t i c u l a r , when t h e p e r -t u r b a t i o n i s z e r o , we i n f e r f r b m t h e c o r r e s p o n d i n g p h a s e - p l a n e t r a j e c t o r i e s t h a t t h e s p i n v e c t o r can p e r f o r m a s y m p t o t i c m o t i o n s o f t h e f i r s t a n d s e c o n d k i n d s and p e r i o d i c m o t i o n s a b o u t p e r m a n e n t a x e s l y i n g i n t h e p r i n c i p a l p l a n e p e r p e n d i c u l a r t o t h e t o r q u e v e c t o r . Some o f t h e r e s u l t s f o r t h i s c a s e were a l s o o b t a i n e d by Gramme1, u s i n g d i f f e r e n t m e t h o d s . I n t h e g e n e r a l c a s e , when t h e p e r t u r b a t i o n i s n o t z e r o , t h e s e m o t i o n s a r e p r e s e r v e d . However, a s e c o n d t y p e o f p e r i o d i c m o t i o n e x i s t s ; i t o c c u r s a b o u t t h e p r i n c i p a l a x i s c o n t a i n i n g t h e t o r q u e v e c t o r , t h e p r i n c i p a l a x i s i t s e l f b e i n g a d i r e c t i o n o f s t a b l e p e r m a n e n t r o t a t i o n . i i i I n S e c t i o n 3 w e c o n s i d e r t h e same p r o b l e m w i t h t h e t o r q u e v e c t o r a c t i n g a l o n g t h e m i d d l e p r i n c i p a l a x i s . U s i n g t h e methods o f t h e p r e v i o u s S e c t i o n , we show t h a t 8 can assume p e r i o d i c m o t i o n s as w e l l as a s y m p t o t i c m o t i o n s o f v a r i o u s k i n d s . The p e r i o d i c m o t i o n s e s t a b l i s h e d , i n t h e s e two S e c t i o n s a r e t h e n c o m p u t e d i n S e c t i o n 4 as power s e r i e s i n a s m a l l p a r a m e t e r . F i n a l l y , i n S e c t i o n 5 t h e m o t i o n o f a s y m m e t r i c r i g i d b ody m o v i n g i n a v i s c o u s medium s u b j e c t t o a t i m e - d e p e n d e n t t o r q u e i s s t u d i e d . I t s m o t i o n i s c o m p a r e d . w i t h t h a t i n a vacuum. We show f i r s t t h a t p , q, and r can be e x p r e s s e d i n t e r m s o f c e r t a i n i n t e g r a l s . F o r t h e s p e c i a l c a s e where t h e s e l f - e x c i t e m e n t i s t i m e - i n d e p e n d e n t a nd f i x e d i n d i r e c t i o n w i t h -i n t h e b o d y , we show t h a t t h e s e i n t e g r a l s can be r e d u c e d t o t h e g e n e r a l i z e d s i n e and c o s i n e i n t e g r a l s . T h e i r v a l u e s can be c o m p u t e d f r o m a s y m p t o t i c and power s e r i e s w h i c h a r e d e v e l o p e d i n t h e same S e c t i o n . The a s y m p t o t i c b e h a v i o r o f t h e s p i n v e c t o r i s t h e n d i s c u s s e d , y i e l d i n g q u a l i t a t i v e r e s u l t s w h i c h a r e sum-m a r i z e d i n t h r e e t h e o r e m s . v i ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o t h a n k h i s a d v i s e r , D r . E. L e i m a n i s , f o r s u g g e s t i n g t h e t o p i c o f t h i s t h e s i s and f o r g u i d a n c e g i v e n t h r o u g h o u t t h e a u t h o r * s g r a d u a t e s t u d i e s and d u r i n g t h e p r e p a r a -t i o n o f t h i s t h e s i s . A l s o , t h e f i n a n c i a l a s s i s t a n c e g i v e n by t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a t h r o u g h i t s summer g r a n t i s a p p r e c i a t e d . i v TABLE OF CONTENTS Page SECTION 1 INTRODUCTION 1 1.1 E u l e r , s Dynamical Equations 1 1.2 The Position of the Rigid Body in Space 3 SECTION 2 TORQUE VECTOR FIXED ALONG THE LARGEST OR SMALLEST PRINCIPAL AXIS 6 2 . 1 Equations of Motion 6 2 . 2 Torque Vector Fixed Along the Largest P r i n c i p a l Axis 7 2 . 3 A Qualitative Discussion of the Motion of the Spin Vector 12 2 . 4 The Motion of the Spin Vector in the Unperturbed Case 13 2 . 5 The Motion of the Spin Vector in the Perturbed Case 24 2.6 Brief Summary and Remarks 43 SECTION 3 TORQUE VECTOR FIXED ALONG THE MIDDLE PRINCIPAL AXIS 45 3 . 1 Equations of Motion and Their Integration 45 3«2 A Qualitative Discussion of the Motion of the Spin Vector 52 3 . 3 The Motion of the Spin Vector in the Unperturbed Case 55 3 . 4 The Motion of the Spin Vector in the Perturbed Case 60 Page 3«5 G e n e r a l Remarks and C o n c l u s i o n s 74 SECTION 4 PERIODIC SOLUTIONS 76 4«1 G e n e r a l C o n s i d e r a t i o n s 76 4»2 P e r i o d i c S o l u t i o n s i n t h e Case o f t h e T o r q u e V e c t o r F i x e d A l o n g t h e L a r g e s t P r i n c i p a l A x i s 79 4.3 P e r i o d i c S o l u t i o n s i n t h e Case o f t h e T o r q u e V e c t o r F i x e d A l o n g t h e M i d d l e P r i n c i p a l A x i s 82 SECTION 5 S E L F - E X C I T E D SYMMETRIC RIGID BODY IN A VISCOUS MEDIUM 87 5.1 E q u a t i o n s o f M o t i o n 87 5.2 The A n g u l a r V e l o c i t y o f t h e R i g i d Body 89 5«3 T i m e - I n d e p e n d e n t T o r q u e V e c t o r F i x e d i n D i r e c t i o n W i t h i n t h e Body 93 5.4 The I n t e g r a l s s i ( x , w) and c i ( x , w) Re w > 0 99 5.5 The A s y m p t o t i c M o t i o n o f t h e S p i n V e c t o r 107 BIBLIOGRAPHY 122 1 SECTION 1 INTRODUCTION 1.1 E u l e r ' s D y n a m i c a l E q u a t i o n s T h i s t h e s i s c o n t a i n s a s t u d y o f t h e m o t i o n o f a r i g i d b ody a b o u t a f i x e d p o i n t s u b j e c t t o a t o r q u e a r i s i n g f r o m i n t e r n a l r e a c t i o n . S u c h a r i g i d body h a s been c a l l e d s e l f - e x c i t e d by Grammel [ 1 1 ] , a n d i t s m o t i o n i s g o v e r n e d by E u l e r ' s d y n a m i c a l e q u a t i o n s . W i t h t h e d e v e l o p m e n t o f i n t e r n a l s e l f - p r o p u l s i o n s y s t e m s , p r o b l e m s o f t h i s t y p e a r e o f g r e a t i n t e r e s t , e s p e c i a l l y i n s p a c e r i g i d - b o d y d y n a m i c s . An e x a m p l e i s t h e p r o b l e m o f r o t a t i o n o f a s p a c e v e h i c l e due t o t h r u s t m i s a l i g n m e n t . I n t h i s i n t r o d u c t o r y S e c t i o n , we s h a l l p r e s e n t t h e e q u a t i o n s o f m o t i o n and some d e f i n i t i o n s and n o t a t i o n s w h i c h w i l l be u s e d t h r o u g h o u t t h e t h e s i s . F o l l o w i n g t h i s , a b r i e f resume o f o u r work w i l l be g i v e n . F i g u r e 1.1 2 In Figure 1.1, let OXYZ and Oxyz be right-hand rectangular coordinate systems fixed respectively in space at the point 0 and in the r i g i d body along the directions of the p r i n c i p a l axes. Suppose p, q, and r are the components of the angular ve l o c i t y vector io along the x, y, and z axes. Then i f A, B, C and M . M , A y Mz are the p r i n c i p a l moments of i n e r t i a and the components of the torque vector 2$ along the same axes, Euler's equations may take the form A'P - (B - C)qr « Mx (1.1) Bq • (A - C)rp = M y Cf - (A - B)pq - Mz Here and throughout our work, dots indicate d i f f e r e n t i a t i o n with respect to the time t. For the po s i t i o n of the torque vector in the body we d i s -tinguish two cases, depending whether 3 i s fi x e d in the body or moves in i t according to some prescribed law. These are c a l l e d respectively self-excitement with fixed or moving direction with-in the body. In addition, nt i s said to be time-dependent or time-independent according as i t s modulus changes with time or not. For a time-independent self-excitement f i x e d in di r e c t i o n within the body, the components M , M , and M become constants. x y z Consequently, i n this special case we have one of the simplest generalizations of the Euler-Po&>»©* case of motion of a heavy r i g i d body. In fact, the problem of motion of a s e l f - e x c i t e d r i g i d body subject to a time-independent torque vector fi x e d in direction within the body and the problem of motion of a heavy 3 r i g i d body i n t h e E u l e r - L a g r a n g e c a s e a r e m a t h e m a t i c a l l y s i m i l a r i n n a t u r e . I n t h e l a t t e r , i t i s known t h a t t h e s o l u t i o n can be r e d u c e d t o q u a d r a t u r e s b e c a u s e a f o u r t h f i r s t i n t e g r a l i n d e p e n d e n t o f t h e t h r e e known ones can be f o u n d . I n t h e f o r m e r , i t f o l l o w s f r o m J a c o b i * s t h e o r y o f t h e l a s t m u l t i p l i e r t h a t t h e s o l u t i o n can a l s o be r e d u c e d t o q u a d r a t u r e s , i f a f i r s t i n t e g r a l i n d e p e n d e n t o f t i s o b t a i n e d . 1.2 The P o s i t i o n , o f t h e R i g i d Body i n S p a c e F i g u r e 1.2 The E u l e r A n g l e s 0, 0, and <f as d e f i n e d i n F i g u r e 1.2 l o c a t e u n i q u e l y t h e p o s i t i o n o f t h e body i n s p a c e . T h e s e a n g l e s a r e known t o be r e l a t e d t o t h e components o f B8 by t h e e q u a t i o n s : p = t> s i n 0 sin<f + 0 cos^ (1.2) q = (5 B i n O c o s f - 0 s i n c f r = 0 cos 0 + f 4 The s o l u t i o n o f ( l . l ) a n d (1.2) c o m p l e t e l y d e t e r m i n e s t h e p o s i -t i o n o f t h e body i n s p a c e a t any t i m e t . I f t h e com p o n e n t s o f m" do n o t c o n t a i n t h e E u l e r a n g l e s e x p l i c i t l y , we can s o l v e ( l . l ) w i t h o u t t a k i n g (1.2) i n t o a c c o u n t . When p, q, and r a r e known as f u n c t i o n s o f t> s o l v i n g (1.2) p r e s e n t s no d i f f i c u l t y s i n c e i t can a l w a y s be r e d u c e d t o q u a d r a t u r e s [ 2 ] . I n t h e f o l l o w i n g , we s h a l l c o n s i d e r f i r s t t h e c a s e i n w h i c h t h e t o r q u e v e c t o r a c t s a l o n g t h e l a r g e s t o r s m a l l e s t p r i n c i p a l i " * i 2 a x i s w i t h component e q u a l t o M • M-0|w| , where M and p. a r e a r b i t r a r y c o n s t a n t s . I t i s shown t h a t E u l e r V s e q u a t i o n s can be i n t e g r a t e d i n t e r m s o f a v a r i a b l e cp. When t h i s i s f o l l o w e d by a q u a d r a t u r e a n d an i n v e r s i o n , t h e n p, q, and r c a n be e x p r e s s e d as f u n c t i o n s o f t . U s i n g t h e m e t h o d o f p h a s e - p l a n e a n a l y s i s , we a r e a b l e t o s t u d y t h e m o t i o n o f to i n t h e l a r g e . I n p a r t i c u l a r , we d i s c u s s a l s o t h e s u b c a s e s : ( i ) \ L Q - 0, ( i i ) | i = 0, M = 0. The f o r m e r i s t h e c a s e c o n s i d e r e d by Gramrael [ 1 1 ] ; t h e l a t t e r i s t h e c a s e o f a f o r c e - f r e e a s y m m e t r i c g r y o s c o p e , t o w h i c h P o i n s o t gave a v e r y e l o q u e n t g e o m e t r i c d i s c u s s i o n . Our t o p o l o g i c a l -a n a l y t i c a l m e t h o d p r o v i d e s a s i m p l e and u n i f i e d a p p r o a c h t o t h e s e p r o b l e m s . I n S e c t i o n 3 we c o n s i d e r t h e same p r o b l e m as i n S e c t i o n 2 bu t w i t h t h e t o r q u e v e c t o r a c t i n g a l o n g t h e m i d d l e p r i n c i p a l a x i s . We o b t a i n r e s u l t s t h a t d i f f e r s i g n i f i c a n t l y f r o m t h e p r e v i o u s c a s e . The p e r i o d i c m o t i o n s w h i c h a r e e s t a b l i s h e d i n t h e s e two S e c t i o n s a r e t h e n c o m p u t e d i n S e c t i o n 4 as power s e r i e s i n a s m a l l p a r a m e t e r . 5 F i n a l l y , i n t h e l a s t S e c t i o n we s t u d y t h e m o t i o n o f a s y m m e t r i c r i g i d body i n a v i s c o u s medium s u b j e c t t o an a r b i t r a r y t i m e - d e p e n d e n t t o r q u e . The c a s e o f a s y m m e t r i c r i g i d body m o v i n g i n a f r i c t i o n l e s s medium s u b j e c t t o a t i m e - i n d e p e n d e n t t o r q u e v e c t o r f i x e d i n d i r e c t i o n w i t h i n t h e body was f i r s t c o n s i d e r e d by Bodewadt [ 2 ] . He showed t h a t t h e g e n e r a l s o l u t i o n o f t h e p r o b l e m c o u l d be e x p r e s s e d i n t e r m s o f t h e F r e s n e l i n t e g r a l s . F o r t h e c o r r e s p o n d i n g s i t u a t i o n i n a v i s c o u s medium, we s h a l l show t h a t t h e p r o b l e m can be s o l v e d by t h e g e n e r a l i z e d t r i g o n o -m e t r i c i n t e g r a l s . T h ese i n t e g r a l s f o r m a c l a s s w h i c h i n c l u d e s t h e F r e s n e l i n t e g r a l s . S e c t i o n 5 t h e n ends w i t h a q u a l i t a t i v e d i s c u s s i o n o f t h e a s y m p t o t i c m o t i o n s o f m. 6 SECTION 2 o TORQUE VECTOR FIX E D ALONG THE LARGEST OR SMALLEST P R I N C I P A L AXIS 2.1 E q u a t i o n s o f M o t i o n I n t h i s S e c t i o n we c o n s i d e r t h e m o t i o n o f a s e l f - e x c i t e d r i g i d body a b o u t a f i x e d p o i n t s u b j e c t t o a t o r q u e v e c t o r 3 l y i n g a l o n g t h e l a r g e s t o r s m a l l e s t p r i n c i p a l a x i s . We s h a l l assume t h a t "A i s t i m e - d e p e n d e n t w i t h component i n t h e c h o s e n a x i s e q u a l t o a c o n s t a n t p l u s a p e r t u r b a t i o n t e r m w h i c h i s p r o -p o r t i o n a l t o t h e s q u a r e o f t h e m o d u l u s o f t h e s p i n v e c t o r CM. T h a t i s , i f M and U.q a r e c o n s t a n t s , we s h a l l t a k e t h e component t o be M + p-0|et)| • Under t h i s a s s u m p t i o n , we s h a l l be a b l e t o o b t a i n e x p l i c i t f o r m u l a s , f o r p, q, and r i n t e r m s o f a v a r i a b l e «p. F o l l o w i n g t h i s , a q u a l i t a t i v e d i s c u s s i o n i s g i v e n o f t h e m o t i o n o f t h e e n d - p o i n t o f 35 i n t h e b o d y - f i x e d t r i h e d r a l , w i t h r e s p e c t t o w h i c h we s h a l l show t h a t t h e e n d - p o i n t can p e r f o r m a v a r i e t y o f m o t i o n s . I n t h e c o u r s e o f t h i s d i s c u s s i o n , p e r m a n e n t d i r e c t i o n s o f r o t a t i o n o f t h e body w i l l a l s o be e s t a b l i s h e d . A l l t h e s e a r e done by g i v i n g a p h a s e - p l a n e a n a l y s i s o f a r e l a t e d d y n a m i c a l s y s t e m . F o r t h e p r e s e n t i t i s c o n v e n i e n t t o t a k e E u l e r ' s e q u a t i o n s i n t h e f o r m p - a q r = ( 2 . 1 ) q + b r p = in y f - . c p q = ra 7 H e r e we h a v e l e t B-C , A-C A-B a - — > b - — ' ° = ~ and M M M X V z m x = A ~ » m y B ' m z = C~ I n a d d i t i o n , w i t h o u t any l o s s o f g e n e r a l i t y , we s h a l l assume t h a t A > B > C so t h a t t h e i n e r t i a numbers a, b, and c a r e p o s i t i v e . 2.2 T o r q u e V e c t o r F i x e d A l o n g t h e L a r g e s t P r i n c i p a l A x i s I f t h e x - a x i s d e n o t e s t h e l a r g e s t p r i n c i p a l a x i s , s y s t e m (2.1) w i t h m = ra =0 becomes y z p - a q r = m + (JU | «i | (2.2) q • b r p = 0 r - c p q = 0 M ^o . . where m = [i - -jj— a r e c o n s t a n t s . I n t r o d u c i n g t h e v a r i a b l e tp by means o f t h e d i f f e r e n t i a l r e l a t i o n dtp = p d t we may w r i t e t h e s e c o n d and t h i r d e q u a t i o n s as ^3 • b r = 0 dtp d r d ^ " c q = 0 The g e n e r a l s o l u t i o n o f t h i s s y s t e m o f e q u a t i o n s c a n i m m e d i a t e l y be p u t i n t h e f o r m 8 (2.3) q = a-jjb c o s y b c tp - ct 2 ( S/F s i n y b c ep r = a i v ^ s i n y b c cp • (X2^ cos<f&c f The a r b i t r a r y c o n s t a n t s Ci^ and o.^ a r e d e t e r m i n e d by t h e i n i t i a l v a l u e s q and r o f q and r a t t = 0, o o I f a nd 0.^ a r e b o t h z e r o , t h e n q = r = 0. A c c o r d i n g l y , t h e r i g i d body r o t a t e s p e r m a n e n t l y a b o u t t h e x - a x i s , f i x e d i n t h e body as w e l l as i n s p a c e , w i t h an a n g u l a r v e l o c i t y p w h i c h s a t i s f i e s t h e e q u a t i o n From t h i s we o b t a i n t h e f o l l o w i n g e x p r e s s i o n s f o r p, d e p e n d i n g on t h e r e l a t i v e m a g n i t u d e s o f m and |i» I n e a c h o f t h e s e e x p r e s -s i o n s , p i s t h e v a l u e o f p at t = 0. $> = ra + |xp 2 ( i ) F o r m > 0, \i. > 0 (2.4) ( i i ) F o r m > 0, [i < 0 • a r c t a n h c o t h (y-rap.t + a r c c o t h p ) » Po > l ? l ( i i i ) F o r m < 0, \i > 0 a r c t a n h 9 (iv) For m < 0, \i < 0 (v) For u, = 0 p = mt • p o (vi) For m = 0, \i t 0 p" 1 - p" 1 - [it i f p £ 0 o r • o p * 0 i f p = 0 o Having disposed of thi s p a r t i c u l a r case, we assume in what follows that at least one of the constants and <x^ i s di f f e r e n t from zero. According to (2.3), this implies that at least one of the i n i t i a l values q Q and TQ i s non-zero. Expressions (2.3) may now be put in the form q = 67b~ cos (Tbctp + a ) (2.5) V r = o^c" sin(^b*b(p + OQ) l~2 2 a2 where 6 = /a, • a„ and a = arctan — are constants which can V 1 2 o be determined from the i n i t i a l values q and r • We proceed now o o to express p as a function of cp. If we substitute expressions (2.5) into the f i r s t equation - . 2 2 2 2 of (2.2) and make use of the fact that |oo| = p + q • r and p • we obtain the second order equation (2.6) * y.6 2[b c o s 2 ( , / E c f + aQ) * c sin 2(ybc<p + CTQ)] 10 t h a t i s s a t i s f i e d by . L e t t i n g and m a k i n g u s e o f t h e i d e n t i t i e s 2 1 + co s 2<px cos f l 2 . 2„ sin cp^ 1-cos 2 ( ^ we can write ( 2 . 6 ) as dt J"c where ' • J i ^ y - I * * * * <>">] • d s in 2((p x + ) 2 °*i = 2 arctan ^(b-c) ] F u r t h e r , p u t ^ 2 - 2 ^ 1 + «1> t = y 2 d Then e q u a t i o n ( 2 . 7 ) r e d u c e s t o F i n a l l y l e t t i n g 1 ( 7\T~ J. U6 2b./bc j ti,6 2c./bc  m l = H ( r a * / b c * ? + ^ 5* 1 d\y 2 2 (2. 9 ) li, -d Cf 2  p l = d*T we obtain the fo l lowing system of f i r s t order equations equivalent 11 t o ( 2 . 8 ) . d p l 2 — = mx • s i n ^ • ( 2 . 1 0 ) d dT " Pi <P2 F u r t h e r m o r e , i t i s c l e a r t h a t 4>2 = 2(ybc"<f + CTo • 0 i > (2.11) m From s y s t e m (2.10) we hav e t h e l i n e a r e q u a t i o n i n p^ » 2 d p l 2 (2.12) - 2 i i l P ^ = 2 m 1 • 2 s i n c p 2 t h e i n t e g r a t i o n o f w h i c h y i e l d s 2 2 ^ l f 2 2 x m l ( 2 . 1 3 ) p x = C 1 e 2 (S^xsincpg • cos<j? 2) - — H e r e i s an a r b i t r a r y c o n s t a n t , whose m a g n i t u d e c a n be f o u n d f r o m t h e i n i t i a l v a l u e s o f p^ a n d ( p 2 # s a y p^Q and <p20* W e h a v e t h e n ( 2.H) Cx - e ^ U P ^ 0 • - - ^ - 2 ( 2 n 1 s i n ( p 2 0 * c o s c f ^ ) • — i*4M-^ * The component p f o l l o w s now f r o m ( 2 . 1 1 ) and ( 2 . 1 3 ) ^ 2 2—,(2^ 1 s i n 0 9 • cosc&J - ~ 1 1 7 < i T < J ^1 I f L J ( f ) d e n o t e s t h e e x p r e s s i o n o b t a i n e d f r o m t h e r i g h t o f (2.15) by r e p l a c i n g c p 2 w i t h t h e f i r s t e q u a t i o n o f ( 2 . 1 1 ) , t h e n (2.16) p - tJUW (2.15) P 2 = ^ c 12 Thus w i t h ( 2 . 5 ) and ( 2 . 1 6 ) we h a v e s u c c e e d e d i n i n t e g r a t i n g E u l e r * s e q u a t i o n s f o r p # q, and r i n t e r r a s o f ^ . I n o r d e r t o e x p r e s s them as f u n c t i o n s o f t f we f i r s t n o t e t h a t , s i n c e d q5 = ± Jxj(f) d t , ( 2 . 1 7 ) t = i f f The i n v e r s i o n o f ( 2 . 1 7 ) p r o v i d e s c|> as a f u n c t i o n o f t . L e t t h i s be d e n o t e d by ( 2 . 1 8 ) Cf- ( f ( t ) Then i n e q u a t i o n s ( 2 . 5 ) and ( 2 , 1 6 ) i f we r e p l a c e (^  by t h e e x p r e s s i o n ( 2 . 1 8 ) , we o b t a i n t h e d e s i r e d r e s u l t . 2.3 A Q u a l i t a t i v e D i s c u s s i o n o f t h e M o t i o n o f t h e S p i n V e c t o r I n S e c t i o n s 2.3 t o 2.5, we a r e c o n c e r n e d w i t h t h e m o t i o n i n t h e l a r g e o f t h e e n d - p o i n t o f t» r e l a t i v e t o t h e b o d y - f i x e d t r i -h e d r a l . F o r b r e v i t y we s h a l l s o m e t i m e s s i m p l y s a y t h e m o t i o n o f a». As we s h a l l l a t e r show, t h i s m o t i o n i s c o m p l e t e l y d e t e r m i n e d by t h e p h a s e - p l a n e t r a j e c t o r i e s o f s y s t e m ( 2 . 1 0 ) . The u s e o f t h e p h a s e - p l a n e m e t h o d a l l o w s us t o d e t e r m i n e t h e t i m e b e h a v i o r o f p, q, and r w i t h o u t h a v i n g t o e v a l u a t e and t o i n v e r t t h e i n t e g r a l i n ( 2 . 1 7 ) . B e c a u s e o f t h e c o m p l e x i t y o f t h e e x p r e s s i o n U(f), i n g e n e r a l t h e s e l a t t e r s t e p s w i l l r e q u i r e some k i n d o f n u m e r i c a l m e t h o d , e x c e p t i n t h e s p e c i a l c a s e s when t h e i n t e g r a l i s an e l l i p -t i c t y p e . From f o r m u l a s ( 2 . 5 ) we n o t e t h a t 13 2 2 2 2 6^b 6 c t h a t i s , t h e p r o j e c t i o n o f t h e e n d - p o i n t o f w o n t o t h e y z p l a n e w i l l a l w a y s l i e on an e l l i p s e w i t h c e n t e r a t t h e o r i g i n and w i t h m a j o r a n d m i n o r s e m i - a x e s a l o n g t h e two p r i n c i p a l a x e s i n t h i s p l a n e . C o n s e q u e n t l y , t h e e n d - p o i n t o f m must a l w a y s move r e l a t i v e t o t h e b o d y - f i x e d t r i h e d r a l on t h e s u r f a c e o f an e l l i p -t i c c y l i n d e r whose e q u a t i o n i s ( 2 . 1 9 ) _JL? + _JL? _ , 2 2 6 b 6 c At t h e same t i m e t h e component p must v a r y a c c o r d i n g t o ( 2 . 6 ) o r t h e e q u i v a l e n t ( 2 . 1 0 ) , w h i c h c o n t a i n s p ^ . B e c a u s e p i s a s c a l a r m u l t i p l e o f p ^ , t h e q u a l i t a t i v e b e h a v i o r o f p i s com-p l e t e l y known w h e n e v e r t h a t o f p^ i s d e t e r m i n e d . I n o r d e r t o see t h e p o s s i b l e v a r i a t i o n s o f p^ and t h u s o f p we s h a l l g i v e a p h a s e - p l a n e a n a l y s i s o f s y s t e m ( 2 . 1 0 ) . From t h i s k n o w l e d g e t h e m o t i o n o f 5) w i l l f o l l o w . We s h a l l d i s t i n g u i s h two c a s e s a c c o r d i n g as \iQ i s z e r o o r n o t . T h e s e w i l l be c a l l e d r e s p e c -t i v e l y t h e u n p e r t u r b e d a n d p e r t u r b e d c a s e s . The f o r m e r w i l l be d i s c u s s e d f i r s t . 2«4 The M o t i o n o f t h e S p i n V e c t o r i n t h e U n p e r t u r b e d C a s e When t h e r i g i d body i s f r e e o f p e r t u r b a t i o n , i t i s t h e n a c t e d upon o n l y by a t o r q u e v e c t o r o f c o n s t a n t m a g n i t u d e f i x e d a l o n g t h e l a r g e s t p r i n c i p a l a x i s . T h i s p r o b l e m was c o n s i d e r e d by Graramel [ 1 1 ] . U s i n g methods d i f f e r e n t f r o m o u r s , he showed t h a t r e l a t i v e t o t h e b o d y - f i x e d t r i h e d r a l i» can h a v e t h r e e types of motion. Two of these he c a l l e d asymptotic, and the other p e r i o d i c . We give here a complete d i s c u s s i o n of t h i s prob-lem by the methods of phase-plane a n a l y s i s . His r e s u l t s w i l l f o l l o w as consequences of the phase-plane t r a j e c t o r i e s of system (2.10) f o r a., = 0. A l s o , such a method allows us to study the p o s s i b l e types of motion that w can perform, when the component of nl takes on d i f f e r e n t values i n p r o p o r t i o n to the i n i t i a l values q , r and the i n e r t i a numbers a, b, and c. Such an o' o ' * a p p a r e n t l y complicated i n t e r r e l a t i o n s h i p turns out to be s u r -p r i s i n g l y simple. We s h a l l a l s o be able to i n d i c a t e i n the phase-plane the regions of bounded and unbounded motions. More-over, the l o c a t i o n s of permanent axes of the body and the con-d i t i o n s f o r the e x i s t e n c e of p e r i o d i c motions of w are e a s i l y e s t a b l i s h e d . The s t a b i l i t y of these two types of motions can a l s o be d i s c u s s e d . From a l l these r e s u l t s , we can l a t e r observe —• 2 the e f f e c t due to the p e r t u r b a t i o n term M-0|n>| . For » 0, system (2.10) becomes dp, dT (2.20) = m, • sincp2 d<P2 d T ~ p i T h i s p a i r of equations r e p r e s e n t s a c o n s e r v a t i v e dynamical system. C l e a r l y , the t r a j e c t o r i e s i n the phase plane are d e f i n e d by dp. m-.+sincp,, C 2 - 2 1 ) - 4 ' - ^ For such a dynamical system an energy i n t e g r a l e x i s t s which can 15 be o b t a i n e d f r o m ( 2 . 2 1 ) by t h e s e p a r a t i o n o f v a r i a b l e s . T h i s g i ves ( 2 . 2 2 ) 2 P l " mlT ° 2 * c ° s C f 5 2 = E where E i s t h e e n e r g y c o n s t a n t . The e q u i l i b r i u m p o s i t i o n s o f t h e d y n a m i c a l s y s t e m c o r r e s p o n d t o t h e s i n g u l a r p o i n t s o f s y s t e m ( 2 . 2 0 ) w h i c h o c c u r a t p o i n t s where t h e n u m e r a t o r and t h e d e n o m i -n a t o r o f t h e e x p r e s s i o n on t h e r i g h t o f ( 2 . 2 1 ) v a n i s h s i m u l -t a n e o u s l y , t h a t i s , a t p o i n t s on t h e ( p 2 ~ a x i s where m 1 + s i n c p 2 = 0 . O b v i o u s l y i f |m^| > 1, no s u c h p o i n t s can o c c u r . As t h e s t r u c t u r e o f t h e p h a s e p l a n e d e p e n d s on t h e v a l u e s o f m^, we f i r s t c o n s i d e r t h e c a s e 0 < |m^| < 1. ( i ) 0 < | m 1 | < 1 The s y s t e m ( 2 . 2 0 ) h a s t h e n an i n f i n i t e number o f s i n g u l a r p o i n t s . They o c c u r i n p a i r s , s p a c e d a t an i n t e r v a l o f 2TT a l o n g t h e c p 2 ~ a x i s . T h e s e p o i n t s c o r r e s p o n d t o t h e p o s i t i o n s o f m a x i -mum and minimum p o t e n t i a l e n e r g y o f t h e d y n a m i c a l s y s t e m . S i n c e t h e s y s t e m i s c o n s e r v a t i v e , a t p o s i t i o n s o f minimum p o t e n t i a l e n e r g y , t h e s e p o i n t s a r e c e n t e r s , w h i l e a t p o s i t i o n s o f maximum p o t e n t i a l e n e r g y t h e y a r e s a d d l e p o i n t s . F i g u r e 2 . 1 shows t h e g r a p h o f t h e p o t e n t i a l e n e r g y V v e r s u s ^  f o r 0 < m^ < 1. F i g u r e 2 . 2 shows t h e p h a s e - p l a n e t r a j e c t o r i e s f o r t h e same v a l u e s o f m^. T h e s e t r a j e c t o r i e s a r e c u r v e s o f c o n s t a n t e n e r g y . T h e y a r e s y m m e t r i c a b o u t t h e q ? 2 ~ a x i s , and e x c e p t a t t h e s i n g u l a r p o i n t s t h e y c r o s s t h i s a x i s o r t h o g o n a l l y . \i1 = 0, 0 < < 1 Figure 2.2 17 The e q u a t i o n s o f t h e s e p a r a t r i c e s , i n d i c a t e d by t h e h e a v y l o o p s i n F i g u r e 2.2, can be o b t a i n e d f r o m t h e c o n d i t i o n t h a t t h e s e c u r v e s p a s s t h r o u g h t h e s a d d l e p o i n t s * I f ( j ^ g d e n o t e s t h e v a l u e o f (p 2 a t t h e s a d d l e p o i n t S, t h e n t h e e q u a t i o n o f t h e s e p a r a t r i x t h a t p a s s e s t h r o u g h S i s g i v e n by (2.23) ?\ = 2 l m i ( f 2 ~ f 2 5 ^ " c o s C f 2 + C O S C P 2 S ^ A l o n g any t r a j e c t o r y G i n t h e r e g i o n s f o r m e d by two c o n -s e c u t i v e s e p a r a t r i c e s , t h e m a g n i t u d e o f p^ t e n d s t o i n f i n i t y . S i n c e p i s a s c a l a r m u l t i p l e o f p^, c l e a r l y p i s a l s o u n b o u n d e d . T h i s i m p l i e s t h a t t h e s p i n v e c t o r t» a p p r o a c h e s t h e p o s i t i v e x - a x i s w i t h i t s m o d u l u s t e n d i n g t o i n f i n i t y , f o r t h e a n g l e b e-twe e n t» and t h e x - a x i s e v e n t u a l l y a p p r o a c h e s z e r o . F o r t h e same r e a s o n , t» t e n d s t o t h e p o s i t i v e x - a x i s f o r any m o t i o n t h a t o r i g i n a t e d on t h e b r a n c h F ( s e e F i g u r e 2.2) o f e a c h s e p a r a t r i x . H o wever, on t h e o t h e r two b r a n c h e s F^ and F^, p ^ and C j ^ t e n d r e s p e c t i v e l y t o z e r o and a f i n i t e v a l u e , s a y cP2s r as t -• +». The s p i n v e c t o r w i t h r e s p e c t t o t h e b o d y - f i x e d t r i h e d r a l a p p r o a c h e s t h e v e c t o r 6yb c o s "J + 6^c s i n H l y i n g i n t h e y z p l a n e . I n s i d e t h e c l o s e d b r a n c h e s o f t h e s e p a r a t r i c e s , we h a v e t h e r e g i o n s o f p e r i o d i c m o t i o n s , s i n c e t h e components p , q and r a l l r e t u r n t o t h e same v a l u e s a f t e r a p e r i o d 18 where t h e l i n e i n t e g r a l i s t a k e n a l o n g t h e c o r r e s p o n d i n g c l o s e d t r a j e c t o r y . N o t e , h o w e v e r , t h a t t h e p e r i o d i c i t y o f i n t h e bo d y -f i x e d t r i h e d r a l d o e s n o t n e c e s s a r i l y mean t h a t t h i s m o t i o n o f t h e r i g i d body i s p e r i o d i c i n s p a c e . I t i s c l e a r t h a t i n s i d e t h e c l o s e d b r a n c h o f e a c h s e p a r a t r i x t h e p e r i o d i c m o t i o n s a r e s t a b l e . M o t i o n s c o r r e s p o n d i n g t o s i n g u l a r p o i n t s r e p r e s e n t p e r m a n e n t r o t a t i o n s o f t h e r i g i d b o d y , s i n c e p, q, and r a r e c o n s t a n t s , and t h e d e r i v a t i v e i s t h u s z e r o . T h u s , t h e r i g i d body r o t a t e s w i t h a c o n s t a n t a n g u l a r v e l o c i t y a b o u t an a x i s f i x e d i n t h e body as w e l l as i n s p a c e . M o r e o v e r , a t t h e s a d d l e p o i n t s t h e p e r m a n e n t r o t a t i o n s a r e u n s t a b l e , w h i l e a t t h e c e n t e r s t h e y a r e s t a b l e . I n a d d i t i o n t o t h e one on t h e x - a x i s , t h e s e a r e t h e o n l y p e r m a n e n t r o t a t i o n s t h a t a r i g i d body s u b j e c t t o s u c h a t o r q u e v e c t o r may h a v e . Grammel [ 1 1 ] c a l l e d a s y m p t o t i c m o t i o n s o f t h e f i r s t k i n d m o t i o n s f o r w h i c h w a p p r o a c h e s t h e x - a x i s w i t h i t s m o d u l u s t e n d i n g t o i n f i n i t y . We s e e t h a t t h e s e m o t i o n s a r e r e p r e s e n t e d by t h e t r a j e c t o r i e s G i n t h e r e g i o n s b o u n d e d by two c o n s e c u t i v e s e p a r a t r i c e s and by t h e t r a j e c t o r i e s F on e a c h s e p a r a t r i x . He c a l l e d a s y m p t o t i c m o t i o n s o f t h e s e c o n d k i n d m o t i o n s f o r w h i c h a» a p p r o a c h e s a f i n i t e v e c t o r f i x e d i n t h e y z p l a n e . O b v i o u s l y t h e s e a r e r e p r e s e n t e d by t h e b r a n c h e s F, and Fg o f e a c h s e p a r a t r i x . P h y s i c a l l y , t h e s e m o t i o n s w i l l r a r e l y o c c u r . I n f a c t , B o g o l i u b o v [3] h a s i n d i c a t e d t h a t t h e p r o b a b i l i t y o f t h e i r o c c u r r e n c e i s z e r o * F i n a l l y , t h e p e r i o d i c m o t i o n s o f w m e n t i o n e d by Grammel a r e i n d i -c a t e d by t h e c l o s e d t r a j e c t o r i e s a b o u t t h e c e n t e r s . 19 F i g u r e 2.4 shows t h e p h a s e - p l a n e t r a j e c t o r i e s f o r -1 < m, < 0, and F i g u r e 2.3 t h e c o r r e s p o n d i n g g r a p h o f p o t e n t i a l e n e r g y . The p o s s i b l e t y p e s o f m o t i o n a r e e s s e n t i a l l y t h e same as b e f o r e , e x c e p t f o r t h e a s y m p t o t i c m o t i o n s o f t h e f i r s t k i n d where now m a p p r o a c h e s t h e n e g a t i v e x - a x i s . H o w e v e r , f o r |m, | = 1, t h e m o t i o n s change c o n s i d e r a b l y . ( i i ) I m J = 1 F o r t h e s e v a l u e s o f ra, t h e s i n g u l a r p o i n t s i n e a c h p a i r c o a l e s c e i n t o one s i n g u l a r p o i n t , w h i c h i s n e i t h e r a s a d d l e p o i n t n o r a c e n t e r . They a r e e q u a l l y s p a c e d a t i n t e r v a l s o f 2v a l o n g t h e ( f ^ - a x i s . The p o t e n t i a l e n e r g y c u r v e w i l l t h u s h a v e n e i t h e r maximum n o r minimum p o i n t s . F i g u r e s 2.5 and 2.6 show t h e p h a s e p l a n e f o r m, = 1 and m, = -1 r e s p e c t i v e l y . A c c o r d i n g t o l f H o s p i t a l , s r u l e t h e s e p a r a t r i c e s a p p r o a c h t h e s i n g u l a r p o i n t s w i t h s l o p e t e n d i n g t o z e r o . S i n c e t h e r e a r e no c l o s e d t r a j e c t o r i e s , no p e r i o d i c m o t i o n s can e x i s t . The s p i n v e c t o r w a p p r o a c h e s a s y m p t o t i c a l l y e i t h e r t h e x - a x i s o r a c o n s t a n t v e c t o r i n t h e y z p l a n e . As t h e o c c u r r e n c e o f t h e l a t t e r m o t i o n h a s a p r o b a b i l i t y o f z e r o , we can s a y t h a t p r a c t i c a l l y 35 a p p r o a c h e s e i t h e r t h e p o s i t i v e o r t h e n e g a t i v e x - a x i s d e p e n d i n g w h e t h e r ra, i s p o s i t i v e o r n e g a t i v e . The p e r m a n e n t r o t a t i o n s a b o u t a x e s i n t h e y z p l a n e w h i c h can s t i l l e x i s t f o r |m,| = 1 a r e c l e a r l y u n s t a b l e . F o r |ra,| > 1, h o w e v e r , no s u c h d i r e c t i o n s can be f o u n d , as shown i n F i g u r e s 2.7 and 2.8. 20 Figure 2.4 21 Figure 2.6 22 ( i i i ) jm1| > 1 Since now there are no s ingular po ints , a l l t r a j e c t o r i e s in the phase plane w i l l go to i n f i n i t y . As a resul t only asymptotic motions of the f i r s t kind can be found. The phase planes are shown in Figures 2.7 and 2.8. Furthermore because m^  = • ' ^ m ' " , 6 aybc 2 i t s absolute value can be made larger than 1 by making 6 , which depends s o l e l y on q Q and r Q , s u f f i c i e n t l y smal l . This f a c t , together with the t o p o l o g i c a l s tructure of the phase plane for |m^| > 1 proves that the d i rec t ion of permanent rotat ion along the x-axis i s s tab le . ( iv) = 0 Since now m^  = ra = 0, we have the case of a force - free asymmetric gyroscope. The motions of c» are described by the polhodes on the Poinsot e l l i p s o i d . However, we want to show here that these motions can also be i n f e r r e d from the corres -ponding phase-plane t r a j e c t o r i e s shown in Figure 2 .9 , thus prov id ing a u n i f i e d approach to a l l these problems. Referr ing to Figure 2.9, we have at a saddle point an unstable ro ta t ion about the middle p r i n c i p a l axis , and at a center a stable permanent rotat ion about the smallest p r i n c i p a l ax i s . For at these points from (2.5) and (2.11) we have res-pec t ive ly p = r = 0, q = 6,/b cos —^~ «= constant and p = q = 0, r = bjc s in —— = constant C l e a r l y these are so lut ions of Euler*s equation for m = 0. Figure 2.9 24 The closed t r a j e c t o r i e s inside the separatr ix correspond to p e r i o d i c motions about the smallest p r i n c i p a l axis as exhibited on the Poinsot e l l i p s o i d , and those curves outside the separatr ix correspond to per iod ic motions about the largest p r i n c i p a l ax is , since on each one of these p , i s per iod i c with a per iod 27T. Thus the separatr ix must have as i t s corresponding part on the Poinsot e l l i p s o i d the separating polhodes. This then completes the discussion on the topo log ica l s tructure of the phase plane of (2.20). We notice that the c r i t i c a l values of m, are 1, 0, and - 1 , since at these values the topo log i ca l structures are changed r a d i c a l l y . 2.5 The Motion of the Spin Vector in the Perturbed Case Turning to the perturbed case, we s h a l l assume without any loss of genera l i ty that |x, = -v < 0. For the pos i t i ve d i r e c t i o n of the largest p r i n c i p a l axis i s a r b i t r a r y ; we may always select i t so that i s negative. The topo log ica l structure of the phase plane is then defined by the equation 2 dp, i , *sincp 0-Vp. (2.24) -r^ = — i-dcp2 p , or i t s equivalent d p l 2 — = m, + . i n c p 2 - vp, U ' 2 5 ) d ? 2 dr p l Thus the s ingular points occur along theCp 2 ~axis at points where m, • s i ncp 2 vanishes. Thus, the i r locat ions are the same as in 25 t h e u n p e r t u r b e d c a s e . I n a d d i t i o n , t h e p h a s e - p l a n e t r a j e c t o r i e s r e m a i n s y m m e t r i c a b o u t t h e c p g - * * * 8 * f ° r w e r e p l a c e p ^ by - p ^ i n ( 2 . 2 4 ) , t h e e q u a t i o n i s i n v a r i a n t . E x c e p t a t t h e s i n g u l a r p o i n t s , t h e y a l s o c r o s s t h e o p g - a x i s o r t h o g o n a l l y . We assume t h a t 0 < m1 < 1 f i r s t . ( i ) 0 < ra2 < 1 Then t h e s i n g u l a r p o i n t s o c c u r a t ( 2 . 2 6 ) ? 2 = a r c s i n ( - m . ) I f 0 d e n o t e s t h e p r i n c i p a l v a l u e o f ( 2 . 2 6 ) , we h a v e T 2 0 • 2kir o r <P2 - 0 • (2k + l ) v where k i s an i n t e g e r . To d e t e r m i n e t h e n a t u r e o f t h e s i n g u l a r p o i n t s a t ^2 = © + 2k7T, we c o n s i d e r t h e e q u a t i o n s o f f i r s t a p p r o x i m a t i o n f o r ( 2 . 2 5 ) a t t h e s e p o i n t s . M a k i n g t h e t r a n s f o r m a t i o n o f c o o r d i n a t e s (0 + 2k7r) we f i n d t h a t o r d e r 2 o r h i g h e r i n 5 . C o n s e q u e n t l y , t h e e q u a t i o n s o f f i r s t a p p r o x i m a t i o n a t t h e s e p o i n t s become dn = ( 2 . 2 7 ) 26 The c h a r a c t e r i s t i c e q u a t i o n o f ( 2 . 2 7 ) i s \ 2 - c o s 0 = 0 -TT w h i c h g i v e s X = t ,/cos 0. B e c a u s e — < 9 < 0 , t h e r o o t s must be r e a l . A c c o r d i n g t o P o i n c a r e ' s c r i t e r i o n , t h e s e p o i n t s a r e s a d d l e p o i n t s . On t h e o t h e r h a n d , * t c f 2 = _ e + (2k + 1 )T t h e e q u a t i o n s o f f i r s t a p p r o x i m a t i o n a r e jft = -§co S e ( 2 . 2 8 ) dT ' The c h a r a c t e r i s t i c e q u a t i o n becomes now A.2 • cos© = 0 w h i c h g i v e s \ = ±,/-cosQ. The r o o t s a r e t h u s p u r e i m a g i n a r y . A c c o r d i n g l y , t h e s e s i n g u l a r p o i n t s a r e e i t h e r f o c i o r c e n t e r s . H o w e v e r , as t h e t r a j e c t o r i e s a r e s y m m e t r i c a b o u t t h e C ^ - a x i s , t h e s i n g u l a r p d i n t s must be c e n t e r s . F o r 0 < ra, < 1 t h e r e a r e t h r e e p o s s i b l e t o p o l o g i c a l s t r u c -t u r e s o f t h e p h a s e p l a n e , d e p e n d i n g on t h e r e l a t i v e v a l u e s o f ra, and V . I n F i g u r e s 2 . 1 0 and 2 . 1 4 we show t h e p h a s e p l a n e s f o r s m a l l and l a r g e v a l u e s o f V r e s p e c t i v e l y . The o t h e r r e m a i n i n g p o s s i b i l i t y o f t h e p h a s e p l a n e i s shown i n F i g u r e 2 . 1 5 . F i g u r e 2 . 1 0 can be e s t a b l i s h e d as f o l l o w s : I n F i g u r e 2 . 12 we f o l l o w t h e t r a j e c t o r y T, t h a t l e a v e s t h e s a d d l e p o i n t S, f o r i n c r e a s i n g v a l u e s o f T» I t must c o n t i n u e t o t h e r i g h t and i n t e r -s e c t s t h e v e r t i c a l l i n e t h r o u g h t h e c e n t e r a t f , , b e c a u s e i n dp, t h e i n t e r v a l ( S 1 # C _ ) , -r—- > 0 f o r s u f f i c i e n t l y s m a l l v a l u e s o f 1 <c aT p , as i n d i c a t e d i n F i g u r e 2 . 1 1 . 27 m. F o r s m a l l v a l u e s o f V < F i g u r e 2 . 1 0 2yi-mY The t r a j e c t o r y T^ t h a t l e a v e s f o r d e c r e a s i n g v a l u e s o f T a l s o i n t e r s e c t s t h e v e r t i c a l t h r o u g h C 9 a t f 0 b e c a u s e now i n t h e dp, * * i n t e r v a l ( c 2 # S2) < 0 . The r e l a t i v e p o s i t i o n s o f f ^ and f 2 d e p e n d on t h e v a l u e o f , f g i s b e l o w f ^ as shown i n F i g u r e 2 . 1 2 . T h i s m V . I f V < can be s e e n by c o m p u t i n g t h e v a l u e s o f p ^ at t h e s e two p o i n t s a t f "2Vf c 2 From e q u a t i o n ( 2 . 1 3 ) t f ^ . 2 2 r  P l = P s l = C s l e and a t f p , o 2 - o 2 C " 2 V*< P l = P s 2 = C s 2 e 1 + 4 V m 2 ( - 2 v s i n c p c + cos<p c ) + — 2 m l - — 2 ( - 2 v ^ n ? c • c o s f c ) • — 28 H e r e C g ^ and C g 2 a r e t h e c o r r e s p o n d i n g v a l u e s o f on t h e t r a j e c t o r i e s t h a t l e a v e t h e s a d d l e p o i n t s and r e s p e c t i v e l y , and c p c i s t h e v a l u e o f c p 2 & t t h e c e n t e r C,,. From e q u a t i o n ( 2 . 1 4 ) , we h a v e 2 V < P s l •4V C . = e s i 2 m l — 2 ( - 2 v s i n ( p s l • cos(p s l) - — and 2vcp £ „r n C = e 8 2 s2 e l ( - 2 V s i n T ° s 2 + C O S ( i P S2 ) " IT 1 + 4 V i n w h i c h cps^ and cpg^ a r e t h e v a l u e s o f (p2 a t S^ and S 2 . S i n c e a t t h e s a d d l e p o i n t s sin c o s f s l = s i n CPs2 = ~ r a l <PB1 = C O S f s2 = 2 2 t h e d i f f e r e n c e p . - p Q can be s i m p l i f i e d t o t h e f o r m S J. S rfC " 2 V f c / 2V(Ps2 2 V C M / m l - 2 V V ^ - ? s l s 2 V A ( l + 4 v 2 ) v From t h i s e q u a t i o n we see t h a t p g ^ - p g 2 < 0 a c c o r d i n g as I—T> m l - 2vyi-ra1 - 0. Thus f 1 i s above f 2 i f v < — v - - , b e l o w m. m1 i f V > — , ^ , and c o i n c i d e w i t h f 2 i f V = — . These t h r e e r e l a t i v e p o s i t i o n s o f f ^ and f 2 g i v e t h r e e d i f f e r e n t t o p o l o g i c a l s t r u c t u r e s o f t h e p h a s e p l a n e . I n F i g u r e 2.12 i f f ^ l i e s above f 2 t h e t r a j e c t o r y T^ c o n -t i n u e s t o t h e r i g h t and c r o s s e s t h e v e r t i c a l t h r o u g h S 2 a t g^. On t h e o t h e r h a n d , t h e t r a j e c t o r y T 2 c o n t i n u e s t o t h e l e f t and must c u t t h e c p 2 ~ a x i s o r t h o g o n a l l y a t some p o i n t g 2 b e t w e e n S^ 29 Figure 2.12 30 and Cg. S i n c e t h e t r a j e c t o r i e s a r e s y m m e t r i c a b o u t t h e <p2~axis, a r e f l e c t i o n a b o u t t h i s a x i s w i l l g i v e t h e c o r r e s p o n d i n g t r a j e c t o r i e s i n t h e l o w e r - h a l f p l a n e . The same p i c t u r e w i l l o c c u r a t o t h e r c o n s e c u t i v e s a d d l e p o i n t s a l o n g t h e <p 2 ~ a x i 8 • Con-s e q u e n t l y , t h e t r a j e c t o r y T, c o n t i n u e s t o t h e r i g h t f r o m t h e p o i n t g , and n e v e r c r o s s e s t h e GOg - 3** 8* E v e n t u a l l y , as e q u a t i o n (2.13) shows, i t w i l l o s c i l l a t e a b o u t t h e l i n e p , = / ™ • The v a l u e s o f p , a l o n g t h e t r a j e c t o r y D w h i c h c u t s t h e v e r t i c a l t h r o u g h S, a t s u f f i c i e n t l y l a r g e v a l u e o f p , d e c r e a s e as we f o l l o w D t o t h e r i g h t . The e x i s t e n c e o f t h e s e p a r a t r i x F on w h i c h P^C^ + 2 i r ^ = p l ^ 2 ^ ° a n * n e n b e e s t a b l i s h e d f r o m c o n -t i n u i t y . H o w e v e r , i t can a l s o be e s t a b l i s h e d f r o m e q u a t i o n (2.13), w h i c h may a l s o be w r i t t e n as 0 - 2 v c P p 0 rai (2.29) P, = C e - y sin(<f> * oJ + ~ where = a r c t a n ^ 3 — ] • *" o r t h e i n i t i a l v a l u e s <^ 20 = ~ C T3' P10 = J "v~~ ' c o n s ' ' : a n ' t C, i s z e r o . I n s u c h a c a s e we may w r i t e (2.30) p - ± / -f - • yL-^ s i n ( q > + 0 ) and t h e e x p r e s s i o n u n d e r t h e r a d i c a l s i g n i s p o s i t i v e f o r a l l m l v a l u e s o f Cp^  i f v < — ' / " " ~2 ' F o r m u l a (2.30) p r o v i d e s t h e n t h e 2 < / 1 - m l e q u a t i o n s o f t h e two s e p a r a t r i c e s F and F, i n F i g u r e 2.10. From t h e above d i s c u s s i o n , t h e g e o m e t r y o f t h e p h a s e p l a n e i n F i g u r e 2.10 i s now c l e a r . The t r a j e c t o r i e s l y i n g above t h e s e p a r a t r i x F w i l l o s c i l l a t e a bout t h e l i n e p , = / — as ^  "* 0 0 ' 31 s i n c e t h e f i r s t t e r m on t h e r i g h t o f (2.29) a p p r o a c h e s z e r o as c|?2 "* On t h e o t h e r h a n d , t h e v a l u e s o f p^ a l o n g t h e t r a j e c -t o r i e s l y i n g b e l o w t h e s e p a r a t r i x F^ w i l l t e n d t o s i n c e now t h e f i r s t t e r m on t h e r i g h t o f (2.29) d o m i n a t e s and d i v e r g e s t o i n f i n i t y as (p 2 -* -<»• B e c a u s e t h e r i g h t h a n d s i d e o f (2.25) i s p e r i o d i c w i t h p e r i o d 2ir i n (p 2, i t i - s a p p r o p r i a t e t o c o n s i d e r t h e c y l i n d r i c a l p h a s e s p a c e o f t h i s s y s t e m . T h i s w i l l be done f o r s m a l l v a l u e s o f V . We t a k e t h e a x i s o f a c y l i n d e r t o be t h e p ^ - a x i s and t h e c u r v i l i n e a r c o o r d i n a t e m e a s u r e d a l o n g t h e a r c o f a r i g h t s e c t i o n o f t h e c y l i n d e r as ( j ? 2 . I n t u i t i v e l y t h e t r a j e c t o r i e s i n t h i s p h a s e s p a c e may be c o n s t r u c t e d by w r a p p i n g t h e p h a s e p l a n e F i g u r e 2.10 a r o u n d t h e c y l i n d e r . The t r a j e c t o r i e s i n t h i s p h a s e s p a c e a r e shown i n F i g u r e 2.13. I n t h i s s p a c e a l l t h e s a d d l e p o i n t s c o i n c i d e i n t o t h e p o i n t S, and t h e c e n t e r s i n t o t h e p o i n t C. The s e p a r a t r i c e s F and F^ a r e now c l o s e d c u r v e s e n c i r c l i n g t h e c y l i n d e r . One b r a n c h o f t h e s e p a r a t r i x t h a t o r i g i n a t e d f r o m t h e s a d d l e p o i n t S s t i l l e n c i r c l e s t h e c e n t r e C. A b o u t C t h e r e i s s t i l l a c o n t i n u u m o f c l o s e d : t r a j e c t o r i e s w h i c h r e p r e s e n t p e r i o d i c m o t i o n s o f u5. H o w e v e r , one o f t h e two o t h e r b r a n c h e s o f t h e s e p a r a t r i x f r o m S now s p i r a l s t o w a r d F as c p 2 -» «>, w h i l e , t h e s e c o n d s p i r a l s t o w a r d F^ as cp2 -* OB. The s e p a r a t r i c e s F and F^ a r e c a l l e d l i m i t c y c l e s o f t h e s e c o n d k i n d [ l ] . - O b v i o u s l y F i s s t a b l e and F^ u n s t a b l e , s i n c e i n a n e i g h b o r h o o d o f F e v e r y t r a j e c t o r y s p i r a l s t o w a r d i t , w h i l e t h e o p p o s i t e i s t r u e n e a r F^. 32 33 The m o t i o n s o f m c a n now be e a s i l y i n f e r r e d f r o m F i g u r e 2.13* A t t h e s i n g u l a r p o i n t s we f i n d t h e p e r m a n e n t r o t a t i o n s o f t h e b o d y . As b e f o r e , i n a n e i g h b o r h o o d o f t h e s t a b l e p e r m a n e n t r o t a t i o n we h a v e a c o n t i n u u m o f s t a b l e p e r i o d i c m o t i o n s . I n a d d i t i o n t o t h e s e p e r i o d i c m o t i o n s t h a t e x i s t e d i n t h e u n p e r -t u r b e d c a s e , we now e s t a b l i s h two new o n e s , n a m e l y , t h o s e r e p r e s e n t e d by t h e l i m i t c y c l e s F and F,. I n t h e s e two m o t i o n s , t h e s p i n v e c t o r », whose e n d - p o i n t a l w a y s l i e s on t h e s u r f a c e o f t h e e l l i p t i c c y l i n d e r , r o t a t e s a b o u t t h e a x i s o f t h i s e l l i p t i c c y l i n d e r and r e t u r n s t o i t s o r i g i n a l p o s i t i o n a f t e r one c o m p l e t e r o t a t i o n . F u r t h e r m o r e , t h e c o m p o n e n t s p, q, and r c a n be e x p r e s s e d i n t e r m s o f t h e J a c o b i a n e l l i p t i c f u n c t i o n s as f o l l o w s . F r o m e x p r e s s i o n s (2.9), (2.11) and (2.30) we f i n d t h a t (2.31) and (2.32) H e r e C p 2 0 i s t h e i n i t i a l v a l u e o f Cp 2 at t = 0, and i f t h e i n i t i a l v a l u e p Q i s p o s i t i v e , t h e p o s i t i v e s i g n i n f r o n t o f t h e i n t e g r a l on t h e r i g h t i s t a k e n . T h i s i n t e g r a l I = <?2 * " 3 ) can be r e d u c e d t o a n o r m a l e l l i p t i c i n t e g r a l o f t h e f i r s t k i n d . F o r m a k i n g u s e o f t h e i d e n t i t y s i n ( ( p 2 • a 3 - i and t h e s u b s t i t u t i o n we may w r i t e ( 2 . 3 3 ) where I = I d© k 2 s i n 2 0 4V m m. B e c a u s e we assume t h a t v < + 2 v 2 I f we l e t F(<f, k) = f , k < 1 . d© 0 ^ k 2 s i n 2 © we o b t a i n ( 2 . 3 4 ) t = + d (ml^ 2 . 7T 2 ^ V Jl+k^J I n v e r t i n g t h e e x p r e s s i o n i n ( 2 . 3 4 ) , we h a v e ( 2 . 3 5 ) <pa - - , 3 - f • 2 am where am d e n o t e s t h e J a c o b i a n a m p l i t u d e f u n c t i o n . I t f o l l o w s f r o m ( 2 . 5 ) , ( 2 . 1 1 ) , ( 2 . 3 1 ) and ( 2 . 3 5 ) t h a t t h e co m p o n e n t s p , q, and r c a n be e x p r e s s e d i n t e r m s o f t h e J a c o b i a n e l l i p t i c f u n c -t i o n s sn and c n . M o r e o v e r , t h e p e r i o d T o f t h e s e p e r i o d i c m o t i o n s can be g i v e n i n t e r m s o f t h e c o m p l e t e e l l i p t i c i n t e g r a l o f t h e f i r s t 35 k i n d v K = r 2 do a •'O y y / l - k 2 8 i n 2 0 ' " F o r c l e a r l y f r o m ( 2 . 3 2 ) 2 r 2 i r d ? 2 T = I / 2 d -0 M a k i n g use o f t h e i d e n t i t y F(n7T 1 <f 2 ' k ^ = 2 n K ± F ^ ? 2 ' n a n i n t e g e r , f r o m t h e t h e o r y o f e l l i p t i c i n t e g r a l s , - we h a v e 2 l v S i n c e F i s a s t a b l e l i m i t c y c l e , t h e p e r i o d i c m o t i o n r e p r e s e n t e d by F i s s t a b l e . H o w e v e r , t h i s i s n o t t h e c a s e f o r t h e m o t i o n r e p r e s e n t e d by F^. Under any s l i g h t d i s t u r b a n c e , t h e e n d - p o i n t o f <8 s p i r a l s away f r o m t h i s m o t i o n ; t h e s p i n v e c t o r a p p r o a c h e s a s y m p t o t i c a l l y e i t h e r t h e n e g a t i v e x - a x i s w i t h i t s m a g n i t u d e t e n d i n g t o i n f i n i t y o r l e s s l i k e l y a c o n s t a n t v e c t o r i n t h e y z p l a n e , o r t h e s p i n v e c t o r c a n t e n d t o assume t h e p e r i o d i c m o t i o n r e p r e s e n t e d by F. S i m i l a r l y , i f t h e s p i n v e c t o r 08 i s d i s t u r b e d f r o m t h e a x i s o f u n s t a b l e p e r m a n e n t r o t a t i o n i n t h e y z p l a n e , i t e i t h e r assumes a p e r i o d i c m o t i o n a b o u t t h e a x i s o f t h e s t a b l e p e r m a n e n t r o t a -t i o n i n t h e y z p l a n e o r t e n d s t o assume t h e p e r i o d i c m o t i o n r e p r e s e n t e d by F. The o t h e r p o s s i b i l i t y o f w r e t u r n i n g t o i t s 36 o r i g i n a l p o s i t i o n i s n o t l i k e l y t o o c c u r . From F i g u r e 2.13 a g a i n we see t h a t , i n v i e w o f e x p r e s s i o n (2.30), i f p Q > / f ~ *• • :.-2- - zg) , (8 assumes a s y m p t o t i c a l l y , i t V W T 1 a p p r o a c h e s t h e n e g a t i v e x - a x i s w i t h i t s m o d u l u s t e n d i n g t o i n f i n i t y . I n o t h e r w o r d s , i f w i n i t i a l l y l i e s i n t h e v i c i n i t y o f t h e n e g a t i v e x - a x i s w i t h s u f f i c i e n t l y l a r g e p r o j e c t i o n on t h i s a x i s , i t c o n t i n u e s t o move s p i r a l l y i n t h i s d i r e c t i o n and e v e n -t u a l l y c o i n c i d e s w i t h t h e x - a x i s . The c o n s t a n t p o s i t i v e s e l f -e x c i t e m e n t f i x e d a l o n g t h e x - a x i s i s u n a b l e t o p u l l «8 ba.ck t o t h e v i c i n i t y o f t h e p o s i t i v e x - a x i s , as i n t h e u n p e r t u r b e d c a s e . F i n a l l y , we n o t e t h a t i f p Q l i e s between t h e above two v a l u e s , t h e n o t h e r t y p e s o f m o t i o n can o c c u r . m l We t u r n now t o t h e c a s e v > — • • v-. As p r e v i o u s l y m e n t i o n e d . t h e t r a j e c t o r y T^ i n d i c a t e d i n F i g u r e 2.14 now c u t s t h e v e r t i c a l t h r o u g h t h e c e n t e r a t f ^ l y i n g b e l o w f 2» From h e r e T^ c o n -t i n u e s t o t h e r i g h t and must c r o s s t h e ( p 2 - a x i s o r t h o g o n a l l y a t some p o i n t b e t w e e n and S 2 » The o t h e r t r a j e c t o r y i n F i g u r e 2.14 c o n t i n u e s t o t h e l e f t a n d i n t e r s e c t s t h e v e r t i c a l t h r o u g h S^. A r e f l e c t i o n a b o u t t h e ( p 2 - a x i s g i v e s t h e i r c o u n t e r p a r t s i n t h e l o w e r - h a l f p l a n e . T h u s , t h e g l o b a l p i c t u r e o f t h e p h a s e - p l a n e t r a j e c t o r i e s i s t h e one g i v e n i n F i g u r e 2.14* I n p a r t i c u l a r we n o t e t h a t t h e a b s e n c e o f t h e s e p a r a t r i c e s F and F^, w h i c h a p p e a r e d i n F i g u r e 2.10, i m p l i e s t h e n o n - e x i s t e n c e o f p e r i o d i c m o t i o n s o f t h e s e c o n d k i n d . F u r t h e r , i f we c o n s i d e r 37 2 - v p , a p p e a r i n g on t h e r i g h t o f ( 2 . 2 5 ) as a moment t e r m due t o some k i n d o f d i s t u r b a n c e s , we c o n c l u d e f r o m a c o m p a r i s o n o f F i g u r e s 2.10 and 2.14 w i t h F i g u r e 2.2 t h a t t h i s moment c a n e s s e n t i a l l y c h a n g e t h e n a t u r e o f m o t i o n . F o r a l a r g e d i s t u r -b a n c e , t h e s p i n v e c t o r , i n s t e a d o f t e n d i n g t o t h e p o s i t i v e x-a x i s , t e n d s now t o t h e n e g a t i v e - h a l f , a c o m p l e t e r e v e r s a l o f d i r e c t i o n • m l I n t h e l i m i t i n g c a s e v = — , t h e p o i n t s f , and f 2 i n 2/1-m, F i g u r e 2.12 c o i n c i d e . M o r e o v e r , t h e c o n s t a n t C, a s s o c i a t e d w i t h t r a j e c t o r i e s T, and i s now z e r o . T h u s , t h e e q u a t i o n o f t h e s e p a r a t r i x shown i n F i g u r e 2.15 c a n be w r i t t e n as (2.36) p 2 = 2 v / l - m 2 [ l - sin(<p 2 + o ^ ) ] On t r a j e c t o r i e s G l y i n g above t h e . s e p a r a t r i x , i n v i e w o f f o r m u l a s ( 2 . 2 9 ) and (2.36), p , e v e n t u a l l y o s c i l l a t e s b e t w e e n t h e v a l u e s 0 and ^ 1 6 (1-m 2) • T h i s means t h a t t h e e n d - p o i n t o f »8 on t h e s u r f a c e o f t h e e l l i p t i c c y l i n d e r w i l l u l t i m a t e l y o s c i l l a t e b e t w e e n t h e p l a n e s x = 0 and x = v^"V 1 6 ( 1 -m 2 ) -H o w e v e r , f o r m o t i o n s r e p r e s e n t e d by t r a j e c t o r i e s G, l y i n g b e l o w t h e s e p a r a t r i x , we f i n d t h a t «5 t e n d s t o t h e n e g a t i v e x-a x i s . F u r t h e r m o r e , we o b s e r v e t h a t i f t» l i e s i n i t i a l l y i n t h e v i c i n i t y o f e i t h e r t h e p o s i t i v e o r t h e ' n e g a t i v e x - a x i s w i t h s u f f i c i e n t l y l a r g e component p ^ , i t w i l l r e m a i n f o r e v e r i n t h e i n i t i a l d i r e c t i o n ; t h a t i s , t h e p o s i t i v e moment o r moment due t o a l a r g e p e r t u r b a t i o n c a n n o t r e v e r s e t h e d i r e c t i o n o f r o t a t i o n , as t h e y w i l l i n s i t u a t i o n s i l l u s t r a t e d i n F i g u r e s 2.10 a n d 2.14» 38 Figure 2.14 Figure 2.15 3 9 Note also that the motion represented by the separatr ix is not a per iod i c motion of the second k ind , since the time of approach to a saddle point i s i n f i n i t e . F ina l ly} we may mention that in this l i m i t i n g case any s l i gh t disturbance of the system gives an essen-t i a l l y d i f f erent topo log ica l p ic ture of the phase plane. ( i i ) -1 < m, < 0 As before we f i n d that the s ingular points occur along the ( p 2 ~ a x i s at points where (2.37) <p2 = a r c s i n ( - m 1 ) . In the same way, i f 0 denotes the p r i n c i p a l value of (2.37), we conclude that the s ingular points at c p 2 = © + 2 k 7 T and (p2 = -© + (2k + l)ir are saddle points and centers re spec t ive ly . The topo log ica l p ic ture of the phase plane is shown in Figure 2.16. This fol lows from the fact that since m, i s non-posit ive the t ra jec tory T^ in Figure 2.16 intersects the v e r t i c a l through C 2 at a point below the one for T 2 » We see that the phase-plane t r a j e c t o r i e s do not d i f f e r e s s e n t i a l l y from those of Figure 2.4, which are given for -1 < m, < 0 and |JL, = 0. Therefore, the motions of 35 for these two cases are of the same types. At th i s po in t , we may mention that because (2.38) B l - <5£ [ m - v&2./bc"(b • c)] 2 m, can become negative i f 6 i s s u f f i c i e n t l y large , even though m representing the constant seIf-excitement i s a very large p o s i -2 t ive number. 1 Since 6 depends so le ly on the values Q and r , we uo see immediately how these i n i t i a l values can affect the motions of Wm ( i i i ) | m 1 | = 1 After the above de ta i l ed d iscuss ions , i t is not d i f f i c u l t to see the topo log ica l s tructures of the phase planes and the corresponding motions of ni for other poss ib le values of m^. For m^  equal to 1 and -1, the phase planes are shown in Figures 2.17 and 2.18 r e s p e c t i v e l y . In each Figure we f ind that every p a i r of saddle point and center in the previous cases coalesces into one s ingular point CS. We note in p a r t i c u l a r the absence of per iod ic motion of the f i r s t k ind . For m^  = 1, we s t i l l have the stable and unstable per iod ic motions of the second k ind . However, these two values of m^  should be considered c r i t i c a l in the sense that any s l ight disturbance of our dynamical system w i l l change the topo log ica l p i c t u r e s , and consequently w i l l a l t er the types of motion. ( iv) | m 1 | > 1 In th i s case since there are no s ingular po ints , there exist no closed t r a j e c t o r i e s in the phase plane. Every t ra jec tory must tend to i n f i n i t y . In Figure 2.19 we observe that for m^  < -1 no per iod ic motion can e x i s t , and u5 tends to the negative x -ax i s , i r r e s p e c t i v e of the i n i t i a l value P Q . For m^  > 1, per iod i c motions of the second kind represented by the separatr ices F and F^ in Figure 2.20 are found. If P q is a value such that i t s corresponding p^-value l i e s above the separatr ix F.. , (5 assumes or tends to assume the per iod ic motion p., » -v < 0, -1 < ra, < 0 F o r m, = 0, t h e s a d d l e p o i n t s and t h e c e n t e r s o c c u r at e v e n and o d d m u l t i p l e s o f W r e s p e c t i v e l y F i g u r e 2.16 42 ^ i , = - V < 0, m , > 1 F i g u r e 2.20 43 r e p r e s e n t e d by F. I f b e l o w , 08 t e n d s t o t h e n e g a t i v e x - a x i s . Note t h a t t h e s e p e r i o d i c m o t i o n s can a l w a y s be e x p r e s s e d i n t e r m s o f t h e J a c o b i a n e l l i p t i c f u n c t i o n s sn and c n . I t f o l l o w s f r o m e x p r e s s i o n (2.38) and F i g u r e s 2.19, 2.20 t h a t t h e p e r m a n e n t r o t a t i o n a l o n g t h e x - a x i s f o r t h e p e r t u r b e d c a s e i s a l s o s t a b l e . F o r i n e x p r e s s i o n (2.38) we c a n make |m^| l a r g e r t h a n 1 by m e r e l y m a k i n g d, whose v a l u e d e p e n d s on q Q and r , s u f f i c i e n t l y s m a l l , o' 2.6 B r i e f Summary and Remarks We h a v e t h u s g i v e n a c o m p l e t e q u a l i t a t i v e d i s c u s s i o n o f t h e m o t i o n o f t h e s p i n v e c t o r ro g e n e r a t e d by a s e l f - e x c i t e d t o r q u e v e c t o r m" a c t i n g a l o n g t h e x - a x i s and h a v i n g a component e q u a l t o M + M-0lro| • We showed t h a t t h e e n d - p o i n t o f «5 moved on t h e s u r -f a c e o f t h e e l l i p t i c c y l i n d e r d e f i n e d by e q u a t i o n (2.19). I t s l o c u s on t h e s u r f a c e c o u l d t a k e v a r i o u s f o r m s , d e p e n d i n g e s s e n -t i a l l y on t h e r e l a t i v e v a l u e s o f m^ and u-^. They were o b t a i n e d on t h e b a s i s o f t h e p h a s e - p l a n e t r a j e c t o r i e s o f s y s t e m (2.10), w h i c h i s s a t i s f i e d by p i r > a s c a l a r m u l t i p l e o f p. We m e n t i o n i n b r i e f t h a t when t r e a t i n g t h e p r o b l e m i n t h i s way we h a v e a l s o c o v e r e d a v a r i e t y o f p r o b l e m s o f s i m i l a r n a t u r e . T h a t i s , t h e r e a r e a number o f p r o b l e m s whose s o l u t i o n s a r e r e -d u c e d u l t i m a t e l y t o t h e s o l u t i o n o f (2.8). F o r e x a m p l e , i f we assume t h a t t h e component o f 3 , f i x e d a l o n g t h e x - a x i s , i s e q u a l 2 2 t o M + H-C(cf + r ) , i t i s e a s i l y s e e n t h a t q and r a r e g i v e n by (2.5) and p s a t i s f i e s an e q u a t i o n w h i c h ca n be r e d u c e d i n f o r m t o (2.8). C o n s e q u e n t l y t h e m o t i o n o f w i n t h i s c a s e i s c l e a r f r o m 44 o u r p r e v i o u s d i s c u s s i o n s . I n a r e c e n t p a p e r , Grammel [13] c o n -s i d e r e d t h e p r o b l e m i n w h i c h t h e t o r q u e v e c t o r a c t e d a l o n g t h e 2 x - a x i s w i t h component M - u p where M and \i a r e p o s i t i v e c o n -s t a n t s . I t i s c l e a r t h a t we have t h e same s i t u a t i o n h e r e as i n t h e p r e c e d i n g e x a m p l e . B a s e d on a s e r i e s o f n u m e r i c a l l y c a l c u -l a t e d c u r v e s , he o b t a i n e d some o f t h e c o n c l u s i o n s t h a t were drawn by us b a s e d on t h e p h a s e - p l a n e t r a j e c t o r i e s o f ( 2 . 2 5 ) . Thus we h a v e s u c c e e d e d i n p u t t i n g h i s r e s u l t s on a more g e n e r a l b a s i s . I n ( 2 . 5 ) and ( 2 . 1 6 ) we h a v e o b t a i n e d e x p l i c i t f o r m u l a s f o r p , q, and r as f u n c t i o n s o f ^ • N e v e r t h e l e s s t h e c o m p l e t e d e s -c r i p t i o n o f t h e m o t i o n o f t h e r i g i d body i n s p a c e r e q u i r e s t h a t t h e p o s i t i o n o f t h e body i n s p a c e be e x p r e s s e d i n t e r m s o f t h e t i m e t . H o w e v e r , h a v i n g p, q, and r as f u n c t i o n s o f cp, t h i s p r o b l e m ca n be r e d u c e d t o q u a d r a t u r e s , as i t i s p o s s i b l e t o e x p r e s s t h e E u l e r a n g l e s , d e t e r m i n i n g t h e p o s i t i o n o f t h e body i n s p a c e , i n t e r m s o f e x p r e s s i o n s t h a t c o n t a i n i n t e g r a l s o f f u n c t i o n s o f <p . S u c h a m e t h o d i s i n d i c a t e d i n B o d e w a d t f s p a p e r [ 2 ] . W i t h t h e i n v e r s i o n o f f o r m u l a ( 2 . 1 7 ) we can t h e n d e t e r m i n e t h e s e a n g l e s f o r any v a l u e o f t . B e c a u s e o f t h e f o r m o f E u l e r ' s e q u a t i o n s , i f t h e t o r q u e v e c t o r l i e s a l o n g t h e s m a l l e s t p r i n c i p a l a x i s , t h e z - a x i s , t h e s o l u t i o n s c a n be o b t a i n e d f r o m t h e p r e c e d i n g o nes by s i m p l y i n t e r -c h a n g i n g t h e a p p r o p r i a t e q u a n t i t i e s t h a t c h a r a c t e r i z e s u c h a m o t i o n . Thus no new p r o b l e m i s p r e s e n t e d i n t h i s p a r t i c u l a r c a s e . H o w e v e r , i f m" a c t s a l o n g t h e m i d d l e p r i n c i p a l a x i s , i . e . t h e y-a x i s , s u c h a m e t h o d w i l l n o t g i v e us t h e s o l u t i o n s . New f u n c t i o n s a r e i n v o l v e d . We w i l l c o n s i d e r t h i s p r o b l e m i n t h e f o l l o w i n g s e c t i o n • 45 SECTION 3 TORQUE VECTOR F I X E D ALONG THE MIDDLE P R I N C I P A L AXIS 3•1 E q u a t i o n s o f M o t i o n and T h e i r I n t e g r a t i o n I f t h e t o r q u e v e c t o r a c t s a l o n g t h e m i d d l e p r i n c i p a l a x i s w i t h component e q u a l t o M + M<0l«i| > E u l e r ' s e q u a t i o n s t h e n h a v e t h e f o r m p - a q r = 0 ( 3 . 1 ) q + b r p = m + u-|m|2 f - c p q = 0 M ^o ^ -where now m = — , u. = — . A g a i n i n t r o d u c i n g a v a r i a b l e cp by means o f t h e d i f f e r e n t i a l e x p r e s s i o n (3-2) d<p - q d t we o b t a i n f r o m t h e f i r s t and t h i r d e q u a t i o n s o f ( 3 . 1 ) t h e s y s t e m d ? " a r = 0 d r „ d ^ " ° P = 0 whose g e n e r a l s o l u t i o n s can be p u t i n t h e f o r m p = &p7a c o s h y a c cp + o^*/* si n h , / a c cp ( 3 - 3 ) r m a^fc sinh,/accp + d^Jc cosh^/ac cp where and a r e a r b i t r a r y c o n s t a n t s d e t e r m i n e d by t h e i n i t i a l v a l u e s p , r , and CP at t = 0 . *o o T o When b o t h and a r e z e r o , e q u a t i o n s ( 3 * 3 ) g i v e p «s r = 0 , and t h e s e c o n d e q u a t i o n o f (3»l) g i v e s 46 ( 3 . 4 ) q - \iq = m In such a case the r i g i d body rotates permanently about the middle p r i n c i p a l ax i s . The so lut ion of ( 3 » 4 ) can be obtained from formulas ( 2 . 4 ) in Section 2 by replac ing p with q, since they both sa t i s fy an i d e n t i c a l equation. Henceforth, we s h a l l assume that at least one of the a's i s d i f f erent from zero; equivalent ly at least one of the i n i t i a l values p Q and r Q i s non-zero. The component q must then sa t i s fy the equation S u b s t i t u t i n g the expressions given by ( 3 « 3 ) in ( 3 « 5 ) and making use of the i d e n t i t i e s ( 3 - 5 ) q = m - brp + u. (p + q + r ) s inh 2 ( A /acf ) cosh 2./ac <P - 1 2 cosh 2 (Jac <p) cosh 2f/ac <P + 1 2 we may write ( 3 . 5 ) in the form ( 3 . 6 ) q - |iq = m + k .s inh 2,/ac cp + k 2 cosh 2,/ac <p where ( 3 . 7 ) m o = m * 2^ a " c ^ a i " a 2^ k, - l i a ^ C a * c) - ( a2 + a 2 } k 2 = ^(a + c ) ( a 2 • a 2 ) - bo^a^/lc" Since d<p = q dt , from ( 3 . 6 ) we have for <p the equation 47 2 2 ( 3 . 8 ) — | - = m D * kjBinh^/accp + k 2 c o s h 2,/ac <p d t \ ' To i n t e g r a t e e q u a t i o n ( 3 » 8 ) and t o f a c i l i t a t e l a t e r d i s c u s s i o n s , we f i r s t r e d u c e ( 3 . 8 ) t o an e q u i v a l e n t s y s t e m o f f i r s t o r d e r e q u a t i o n s by c h a n g i n g t h e d e p e n d e n t and i n d e p e n d e n t v a r i a b l e s . F o r t h i s p u r p o s e , we c o n s i d e r t h r e e c a s e s , d e p e n d i n g on t h e r e l a t i v e v a l u e s o f k^ and k 2 « Case 1. I k J > | k g | Under t h i s a s s u m p t i o n , we h a v e k 1 s i n h 2k/accp+ k 2 c o s h 2,/accp = e ^ ^ i n h (2,/ac (f) + a 1) where d i = M - 4 k2 cr, = a r c t a n h -— 1 k x £^ = sgn k^ E q u a t i o n ( 3 « & ) assumes t h e n t h e f o r m ( 3 . 9 ) ^ - ^(dT) 2 = % + £ 1 d 1 s i n h ( 2 y * a ^ f + a±) L e t A[ 2o\/ac Then ( 3 . 9 ) becomes d T 2 2 7 ^ F u r t h e r p u t t i n g (3 .10) - J - - 1 M ^ 2 - • V i „ h ? 2 48 2 y a c ra o ( 3 . 1 1 ) ra, = d(?2 d T = q i we o b t a i n t h e f o l l o w i n g s y s t e m o f f i r s t o r d e r e q u a t i o n s e q u i v a -l e n t t o ( 3 - 1 0 ) d q l 2 — = m, > e l S i n h c p 2 * {,1a1 ( 3 ' 1 2 ) d<P2 dr~ = q l O b v i o u s l y we h a v e <P = ( ? 2 " a,) ( 3 . 1 3 ) _ r - d j q = j _ Q i 2 From ( 3 . 1 2 ) t h e f i r s t o r d e r l i n e a r e q u a t i o n i n q , f o l l o w s . d q l 2 ( 3 . 1 4 ) -5— - 2 M- 1q 1 - 2 m , + 2 e l S i n h < p 2 T h i s e q u a t i o n c a n be i n t e g r a t e d t o o b t a i n 2\i^ 2C m ( 3 . 1 5 ) " 1 c i e • | ( 2 ^ 1 s i n h C p 2 + cosh(p 2) - ^ where t h e c o n s t a n t c, i s d e t e r m i n e d by t h e i n i t i a l v a l u e s °;,Q# *?20 a t t = 0 . I n t h e i n t e g r a t i o n we assume t h a t £ S o l u t i o n c o n t r a r y t o t h i s a s s u m p t i o n c a n be c o n s i d e r e d as a l i m i t i n g c a s e o f ( 3 . 1 5 ) . 49 Case 2. j k 1 | < | k 2 j In t h i s c a s e , we hav e k ^ i n h 2Jacd{+ k g c o s h 2Jac(p = £ 2 d 2 c o s h (2jTccp + a2) y'~2 2~ k l k 2 ~ k ^ , a 2 = a r c t a n h — , and e 2 = sgn k 2« As b e f o r e , w i t h cp = 2,/acCp + a 2 and t = 1 T 1 e q u a t i o n (3.8) V 2d 2 yac t a k e s t h e f o r m d2Cp / d f o \ 2 ™ C o r r e s p o n d i n g t o e q u a t i o n s (3»12) i n t h e p r e c e d i n g c a s e , we hav e now t h e f o l l o w i n g s y s t e m e q u i v a l e n t t o (3.16): d q l 2 — = m 2 + e 2 c o s h c p 2 * ^ ( 3 ' 1 7 ) d<p2 dT" = q i m H e r e u., = ^ , = , M o r e o v e r , we ha v e 2,/ac ^ d2 2,yac 1 2 2 d2 2/ac 2 E q u a t i o n s (3.17) l e a d t o t h e f i r s t o r d e r l i n e a r e q u a t i o n i n q^ dq-(3.IS) ^ - 2 i i x q2 » 2m2 + 2e 2coshcp 2 '2 f r o m w h i c h we o b t a i n 2u.,cp 2e ? m (3.19) q f = c e • ^ r ( 2 l i coshCO + s i n h c p 2 ) - ~ where c^ i s t h e c o n s t a n t o f i n t e g r a t i o n . 50 Case 3 . J k x j = | kg | * 0 F i n a l l y we may w r i t e k l B i n h 2/acCp • k g C o s h 2 y a c ( p = ( s g n k,) | k , | e 2 ^ 8 0 *P i f k , = k 2 = - ( s g n k 1 ) | k 1 | e " 2 ' / a c ' P i f k , = - k g E q u a t i o n (3«&) t h e n becomes r e s p e c t i v e l y ( 3 . 2 0 ) 4 ; . „ . o . ( s g n k l ) | k l | . v « q > and ( 3 . 2 1 ) - n ( f ) 2 - m o - ( s g n k^lkjI.-V"? I f we l e t CP 0 = 2,/ac@, t = , T ===== , e q u a t i o n s ( 3 . 2 0 ) and V 2 i k i i ^ ( 3 . 2 1 ) b e c o m e and ( 3 . 2 3 ) ^ - - ^ f — 2 ) = l A - ( B P k j e " ^ d T 2 2 y a c ^ d T / ' "1 U m o d < ?2 F u r t h e r p u t t i n g u., = ^ m^ = -J-J;—j- , - p — = q , f we have r e s p e c t i v e l y t h e f o l l o w i n g s y s t e m s o f f i r s t o r d e r e q u a t i o n s e q u i v a l e n t t o ( 3 - 2 2 ) and ( 3 . 2 3 ) . d q l ? 2 2 m^ + ( s g n k , ) e + ^ i q i ( 3 . 2 4 ) ^ IT = q i 51 d q l , -?2 2 -£~T = - ( s g n k 1 ) e + ^ l q l (3.25) d T = q i 2 k. In e a c h c a s e , = - " - . q = / 11 ^  q, > 2 y a c * V 2«/ a c From s y s t e m (3.24) we o b t a i n t h e f i r s t o r d e r e q u a t i o n (3.26) ^1 - 2u.,q 2 = 2m • ( s g n k ) 2 e ^ 2  d t 2 3 t h e i n t e g r a t i o n o f w h i c h y i e l d s 0 2 ^ 1 < P 9 0 ^ 2 m* (3.27) 4 - c x e • ( s g n k,) f ^ - - ^ I n t h e same way we have f r o m (3«25) (3.28) q 2 - C l e • ( s g n k,) - ^ I n b o t h e q u a t i o n s c^ i s an a r b i t r a r y c o n s t a n t . A c c o r d i n g l y , f r o m f o r m u l a s (3.15), (3.19), (3*27) o r (3.28) we may e x p r e s s q as a f u n c t i o n o f Cp, s i n c e i n e a c h c a s e q i s a s c a l a r m u l t i p l e o f q^ a n d c p 2 i s a l i n e a r f u n c t i o n o f cp . L e t t h i s be d e n o t e d s y m b o l i c a l l y by (3-29) q = t V(cp) Then t h e i n v e r s i o n o f t h e i n t e g r a l (3.30) t - , -flfa o b t a i n e d f r o m (3.2) a l l o w s us t o e x p r e s s cp as a f u n c t i o n o f t ; t h a t i s . 52 ( 3.3D <p-<p(t) This expression is then the so lut ion of ( 3 . 8 ) . -Putt ing ( 3 « 3 l ) in formulas ( 3 . 3 ) and ( 3 - 2 9 ) # we obtain p, q and r as functions of t , 3 . 2 A Qua l i ta t ive Discussion of the Motion of the Spin Vector In Sections 3 « 2 to 3 « 4 we s h a l l give a q u a l i t a t i v e d i scus-sion of motions of the end-point of 3> with respect to the body-f ixed t r i h e d r a l , using the phase-plane method. It w i l l be shown that these motions are e s s e n t i a l l y d i f ferent from the ones when the torque vector acts along the largest or smallest p r i n c i p a l axi s. In the f i r s t place we note from equations ( 3 . 3 ) that ( 3 . 3 2 ) or ( 3 . 3 3 ) This means that the project ion of the end-point of t» onto the xz plane always l i e s on one of the conjugate hyperbolas: a 2 i f |ot11 > | a 2 | a* i f l a j < | a 2 | ( 3 . 3 4 ) ^ - ^ - a l and ( 3 . 3 5 ) c a 2 53 These h y p e r b o l a s a r e i l l u s t r a t e d g r a p h i c a l l y i n F i g u r e 3«1 i n w h i c h 08 i s t h e o r t h o g o n a l p r o j e c t i o n o f (?) o n t o t h e x z p l a n e . X z The m o t i o n o f 35 i n t h e b o d y - f i x e d t r i h e d r a l must be s u c h t h a t i t s e n d - p o i n t a l w a y s l i e s on t h e s u r f a c e o f one o f t h e h y p e r b o l i c c y l i n d e r s d e f i n e d by (3.34) and (3-35). I f l a i l = l 0 ^ ! ' e q u a t i o n (3.32) d e g e n e r a t e s i n t o t h e e q u a t i o n s (3.36) • = 0 and (3-37) - f - - ~ = 0 The c o n j u g a t e h y p e r b o l a s d e g e n e r a t e i n t o t h e s t r a i g h t l i n e s ( s e e F i g u r e 3.2) (3-38) z = x and (3.39) * - s * I n s u c h a c a s e , t h e v e c t o r w w i l l a l w a y s l i e a l o n g one o f x z t h e s e s t r a i g h t l i n e s , w h i l e t h e s p i n v e c t o r t8 w i l l move i n s u c h a way t h a t i t s e n d - p o i n t w i l l r e m a i n f o r e v e r i n one o f t h e two p l a n e s , s a y I and I I , d e f i n e d r e s p e c t i v e l y by (3.38) and (3«39)« C l e a r l y t h e s e p l a n e s a r e t h e two p l a n e s o f t h e s e p a r a t i n g p o l -h o d e s o f a f o r c e f r e e a s y m m e t r i c g y r o s c o p e . F u r t h e r m o r e , b e c a u s e f o r m u l a s (3«3) can now be w r i t t e n as 54 F i g u r e 3.2 e (3.40) P -55 = a iy c* e ^ i f a 1 = a. and , - jac cp (3.41) p = V a e * r = -a 1 ( / c T e~J a cT, i f a x =-a 2 t h e s p i n v e c t o r i» w i l l r e m a i n a c c o r d i n g t o t h e a l g e b r a i c s i g n o f i n e i t h e r t h e u p p e r o r t h e l o w e r - h a l f o f p l a n e I i f ttl = a 2 ' a n d * n e * t h e r t h e u p p e r o r t h e l o w e r - h a l f o f p l a n e I I i f a± - - a 2 . The l o c u s o f t h e e n d - p o i n t o f t» on t h e s e s u r f a c e s o r p l a n e s can t a k e a v a r i e t y o f f o r m s , d e t e r m i n e d by e q u a t i o n (3«8)» We c o n s i d e r f i r s t t h e u n p e r t u r b e d c a s e U - Q = 0, t h a t i s , t h e c a s e where t h e r i g i d body i s s u b j e c t t o a c o n s t a n t s e l f - e x c i t e m e n t f i x e d a l o n g t h e m i d d l e p r i n c i p a l a x i s . We s h a l l show t h a t t h e l o c u s i s a c l o s e d c u r v e on t h e c y l i n d e r s , and w i t h one e x c e p t i o n i s a l s o a c l o s e d c u r v e on t h e two p l a n e s o f t h e s e p a r a t i n g p o l -h o d e s . 3 »3 The M o t i o n o f t h e S p i n V e c t o r i n t h e U n p e r t u r b e d Case In t h i s c a s e , e x p r e s s i o n s (3»7) f o r m Q, k^, and k 2 become m = m o k = ( a 2 2 s K l 2 ^ 1 2J k 2 = -bJTc o.xa2 I f ^ l 0 ^ ! * t h e n |k^| > | k 2 1, and sgn k^ i s n e g a t i v e . E q u a -t i o n (3»8) can t h e n be r e d u c e d t o (3«12), w h i c h now assumes t h e f o r m 56 dq1 — = m, - sinhcp ( 3 ' 4 2 ) D The q u a l i t a t i v e behaviors of q and cp w i l l be known as soon as those of q , andcpg a r e determined, since they are re lated by expressions ( 3 » 1 3 ) « The s ingular point of system ( 3 « 4 2 ) occurs on the cpg -axis at the point where C p ^ = arcsinh m, Since equations ( 3 . 4 2 ) represent a conservative dynamical system, an energy i n t e g r a l \ q\ * ( - m f ? 2 + c o s h c ? 2 ^ = E e x i s t s . Here E is the energy constant, and the expression v = - r a 1 c p 2 * c o s h °f 2 i s the p o t e n t i a l energy of the system. Since c $ 2 = " r a l + s i n h c P 2 2 d V U = C ° 8 h C P 2 dcp2 the p o t e n t i a l energy curve is concave upward. The s ingular point i s thus a center with a l l other t r a j e c t o r i e s in the phase plane surrounding i t , as shown in Figure 3 « 3 « Consequently, the locus of the end-point of 35 on the surface of the corresponding cy l inder is always a closed curve. In other words, 35 assumes a per iod ic motion. The s ingular point represents m, < 0 ra, = 0 ra, > 0 Figure 3.3 a d i r e c t i o n of permanent rotat ion in each c y l i n d e r . Further-more, th i s Figure also indicates that the permanent rotat ion along the middle p r i n c i p a l axis i s unstable as in the force-free case. If |a,| = I A2I ' then w moves in one of the two planes of the separating polhodes, and equation (3.8) can be reduced to - e i f a, - a 2 • e i f a, - -a 2 The s ingular points of the above systems occur respect ive ly along the Cp--axis at points where ( 3 . 4 3 ) or (3.44) d q , dT d ^ dT = q . d q . dr" = m. d ^ dT 58 ( 3 . 4 5 ) m3 - e * = 0 or ( 3 . 4 6 ) m3 • e 2 = 0 These two systems have respect ive ly the energy integrals ( 3 . 4 7 ) 2 q l ~ m3$2 * e 2 = E or 1 2 „ . ~^2 ( 3 . 4 8 ) 2 q l " m 3 ^ 2 * e E In Figures 3 » 4 and 3 * 5 are given the phase-plane t r a j e c t o r i e s for = and = _ a 2 " These can be es tabl i shed with reference to the corresponding p o t e n t i a l energy curves or to equations ( 3 . 4 5 ) to ( 3 . 4 8 ) . In subcases (d) and (e) of each Figure are given the phase-plane t r a j e c t o r i e s when the systems have no s ingular po int s . N a t u r a l l y , in such s i t u a t i o n s , every t ra jec tory tends to i n f i n i t y . In subcases (e), both | q 1 | and |<f2| t end to i n f i n i t y . Conse-quently , | q | increases i n d e f i n i t e l y , and at the same time, accor-ding to ( 3 » 4 C ) and ( 3 . 4 I ) , P and r approach zero. The spin vector thus tends toward the negative or the pos i t ive y-axis with [u5j -» <*>„ On the other hand, in subcases (d) o5 also tends to the y - a x i s , but with a f i n i t e modulus, since now the t r a j e c t o r i e s in these Figures approach asymptotical ly hor i zonta l l ines in the phase plane . In the l a t t e r two subcases the motions of «$ are also well-known from the Poinsot e l l i p s o i d . From Figures 3 " 4 and 3 . 5 we see that ^ in general assumes p e r i o d i c motions in the planes of the separating polhodes, except when the systems ( 3 « 4 3 ) and ( 3 » 4 4 ) have no s ingular po int s . 59 (d) m0 = 0 force - free (e) m < 0 F i g u r e 3.4 a = a 6 0 3.4 The Motion of the Spin Vector in the Perturbed Case Without any loss of genera l i ty , we assume that [i i s nega-t i v e , i . e . H = - | p.J. Furthermore, we s h a l l f i r s t make the addi t iona l assumption that | a , j ^ | | . This implies that the end-point of the spin vector moves on the surface of one of the conjugate hyperbol ic c y l i n d e r s . Case 1. | a , | ^ | a 2 | The constants k, and kg become now k 1 = - ^ i ( a + c ) 2 a 1 a 2 - ^ ( a 2 + a 2 ) (3-49) „ k 2 - • c ) ( a 2 • a 2 ) - ^ S £ 2a1u2 In Figures 3*6 and 3*7 we give respect ive ly the graphs of k , and k 2 versus j |JU | for a , and CL^ having the same and opposite s igns . It fol lows from the dif ference (3.50) k , - k 2 = | [ In-1 (a • c) - by'a^J(a 1 - a , , ) 2 that in each Figure the two s tra ight l ines intersect at the point l y i n g below the | t i . | -axis . Moreover, from these two Figues together with (3«50) and the sum (3.51) k, • k 2 = - ^ [ | n | ( a + c) + bjkll(a1 * a 2 ) 2 we see that i f < [i.^, then | k , | > | k21 and sgn k, negative, and i f > then | k 2 j > | k , | and sgn kg negat ive . 61 sgn a , = sgn sgn a , = -sgn Figure 3.6 Figure 3.7 Let us suppose f i r s t that |u,| < u,^. Equation (3 .8) , which governs the behavior of q and op, can be reduced to (3-12), which now assumes the form dq. (3.52) where JA, (3.53) d r 1 = m l " s i n h ( P 2 " V q l d < P 2 _ dT ~ q l = - V = ~ 1M-1 . j n a d d i t i o n , t h e c o n s t a n t 2,/ac is always less than 1/2. For i f r , and denote respect ive ly A A the ra t ios of moments of i n e r t i a and ^ , then according to the d e f i n i t i o n s of the i n e r t i a numbers a, b, and c we have 2 b 1 r l 2(a+c) 2 r 1 ( r 2 + l ) - r 2 " 62 Because of the assumption A > B > C, i t follows that > 1, r 2 = r^ + h , h > 0, and 2 1 r l 1 2 ( a + c ) 2 2A, , -n 2 r 1 + h ( r 1 - i ; The so lut ion q^ in terms of Cp 2 i s given in equation (3*15) which can be written as ( 3 - 5 4 ) qf = c e * - * . cosh(f * a J + ~ where = arctanh (-2v) D The s ingular point of the system occurs on the cp 2~axis at ^2 = arcsinh m^« Making the l inear transformation n = q x we f i n d that the equations of f i r s t approximation at the point assume the form do d? = "? c o s h <P 2e ( 3 . 5 5 ) ^ d T 9 The c h a r a c t e r i s t i c equation is A. 2 + cosh c p 2 e = 0 from which we obtain the c h a r a c t e r i s t i c roots A. = + ^  -cosh Cp 2 e . Since the t r a j e c t o r i e s are symmetric about the cp 2 ~axis, the s ingular point is a center . A l l other t r a j e c t o r i e s in the phase plane are closed curves surrounding th i s point as shown in Figure 3 " 8 . For instance, in Figure 3 . 8 (a ) i f we fol low the t ra jec tory through the point f , i t w i l l continue to the r ight as T increases and must intersect 63 at f^ the v e r t i c a l through the s ingular p o i n t . This t ra jec tory cannot cross the (p 2 ~axis between the points f Q and f^, because - d q x to the l e f t of the v e r t i c a l , y > 0 for s u f f i c i e n t l y small values of q^. From f^ i t continues to the r ight and cuts the ¥ 2 ~ a x i s d q l orthogonally at g^, as to the r ight of the v e r t i c a l , i s always less than zero. If we now fol low the same t ra jec tory from f for decreasing T , i t can e i ther cut the cp^-axis at g,>, as shown, or continue forever to the l e f t without cross ing the cp 2~axis. However, i f v < the l a t t e r cannot happen, for then the r ight side of ( 3 « 5 4 ) w i l l assume an a r b i t r a r i l y large negative value, while the l e f t side can assume only non-negative values. Since the t r a j e c t o r i e s are symmetric about the <P 2-axis, a r e f l e c t i o n about th i s axis gives then the c losed t ra jec tory through f Q . Thus we see that per iod ic motions of UD are preserved i f i s s u f f i c i e n t l y smal l . On the other hand i f |n| > we s h a l l see that per iod i c motions can ex i s t only for small values of q . Since now o | kg | > | k j j , equation ( 3 » & ) can be reduced to d q l 2 — = m 2 - cosh f 2 - vq x ( 3 . 5 6 ) dcp 2 dr~ = q l The ..solution q^ in terms of (p2 i s given by formula ( 3 « 1 9 ) with u.^  replaced by - v and £ 2 = sgn k 2 negative. The s ingular points of ( 3 « 5 6 ) occur along the f 2 ~ a x i s at points where 64 - cosh cp g = 0 . For > 1 there are two such po in t s , one on each side of the o r i g i n and equidistant from i t . Their nature can be deter-mined from the equations of f i r s t approximation " - ; = i n l " P 2 i d T ' where cf^ ( i = 1 , 2) denotes the values of cp2 a t the s ingular po in t s , and £ = <P2 - cp 2 i ' ^ = q^. From the c h a r a c t e r i s t i c equation of ( 3 . 5 7 ) we f i n d that A. = ± y - s i n h c p , ^ . Hence at ^ ®' the s ingular point i s a center, and at c p 2 ^ < 0 a saddle p o i n t . The phase-plane t r a j e c t o r i e s are shown in Figure 3«9» These are es tabl i shed from the above resul t s and equation ( 3 « 1 9 ) ' On the t r a j e c t o r i e s G^ and G2, q^ -• ± 0 0 a s <p 2 ~*~a,B Because the hyperbol ic sine and hyperbol ic cosine functions are unbounded, on any of the t r a j e c t o r i e s in Figure 3«9 that tend to i n f i n i t y , | p | , | r | , and | q | -• 0 0 . On the other hand, in any motion that or ig inated on the t r a j e c t o r i e s G^ and F , a> tends to a f i n i t e vector in the xz plane. However, such motions are c l e a r l y un-s table , since under s l ight disturbance, eo w i l l assume e i ther a per iod i c motion about a permanent axis of ro ta t ion or an asymp-t o t i c motion, say of the t h i r d k ind , in which | p | , |<j|# | r | •* <», and t» tends to coincide with one of the two asymptotes of the conjugate hyperbolas . In Figures 3 . 1 0 and 3 . 1 1 are given the phase planes for = 1 and ra^ < 1 . (See page 68.) 65 m 2 > 1 F i g u r e 3.9 66 I n t h e c a s e ra^ = 1 t h e s a d d l e p o i n t and t h e c e n t e r c o a l e s c e i n t o a s i n g l e p o i n t at t h e o r i g i n . 35 can assume o n l y a s y m p t o t i c m o t i o n s o f t h e s e c o n d and t h i r d k i n d s . I f m 2 < 1, no s i n g u l a r p o i n t f o r s y s t e m (3«56) can o c c u r . The s p i n v e c t o r m can now assume o n l y a s y m p t o t i c m o t i o n o f t h e t h i r d k i n d . I n t h e l i m i t i n g c a s e = \i, , we h a v e |k,| = | k g | and sgn k, and sgn kg b o t h n e g a t i v e . S i n c e e q u a t i o n (3«8) can now be r e d u c e d t o (3.58), t h e p h a s e - p l a n e t r a j e c t o r i e s a r e shown i n F i g u r e s 3.12 and 3.13 d e p e n d i n g w h e t h e r m^ i s g r e a t e r t h a n z e r o o r n o n - p o s i t i v e . The p o s s i b l e m o t i o n s o f at f o l l o w e a s i l y f r o m t h e s e F i g u r e s . C a s e 2. (0..J » \<x \ L e t us assume now t h a t t h e s p i n v e c t o r 35 l i e s i n i t i a l l y on one o f t h e two p l a n e s o f t h e s e p a r a t i n g p o l h o d e s . T h i s means P o ra~ t h a t — = ±1— and a c c o r d i n g l y a , = + a g . F u r t h e r m o r e , as n o t e d o i n S e c t i o n 3«2, t» w i l l a l w a y s r e m a i n i n one o f t h e two p l a n e s . E q u a t i o n (3.8) c a n now be r e d u c e d t o (3.24) o r (3.25). I f we r e p l a c e [i, by - V , t h e t r a j e c t o r i e s i n t h e p h a s e p l a n e a r e t h e n d e f i n e d by t h e e q u a t i o n 2^ 2 dq. m +(sgn k..)e - V q . (3-58) — ± = -2 i - -1 i f a , - a 2 o r -<P2 2 dq. m - ( s g n k . ) e - V q . . (3.59)" ^ ± - -2 i i f a , - - a 2 We s h a l l s e e t h a t t h e p h a s e - p l a n e t r a j e c t o r i e s d e f i n e d by t h e s e two e q u a t i o n s a r e e s s e n t i a l l y d i f f e r e n t . Thus t h e b e h a v i o r o f 35 i n p l a n e s I and I I w i l l be d i f f e r e n t . 67 F i r s t l e t us assume that 0.^  = a^. Then tS always moves in plane I . According to (3••7), sgn k^ is negative. Solut ions of (3.58) can be written as -2v f f m (3.60) q 2 - C l e ' - T ^ e ^ ^ . Further , equation (3«58) has a s ingular point on the <p2~ a x i s a t = In m_ 1 2e 3 provided that m^  > 0. Le t t ing § = cp2 _ <P2e a n d ^ = q l ' W e obtain at the s ingular point the equations of f i r s t approxima-t ion do -d T ' 2 The corresponding c h a r a c t e r i s t i c equation is A, + m^  = 0, which gives the roots A, = • y-m^. Since m^  > 0 and the t r a j e c t o r i e s are symmetric about the cPg-axis as indicated by (3.58), the s ingular point i s a center. The phase-plane t r a j e c t o r i e s for m^  > 0 and ra^ < 0 are shown in Figures 3»12 and 3«13 respec-t i v e l y . In Figure 3.12 the separatr ix F is determined by the fact that in equation (3»60) i t is poss ible to select a sui table set of i n i t i a l values a n d q 10 s o t h a t t n e constant of in tegra-t ion c^ i s zero. Thus the equation of the separatr ix becomes 2 ^ _ 2 _ *2 q l " v " 1+2V e m2 which shows that q^ -• ± I — as cp^  "* ~ 0 0 ' 68 F i g u r e 3.10 F i g u r e 3.11 m 3 > 0 t n>3 < 0 F i g u r e 3.12 F i g u r e 3.13 69 In the unperturbed case n = 0, the spin vector uo of a force - free asymmetric gyroscope has the tendency to coincide with the negative y - a x i s . However, as observed in Figure 3«4# any a r b i t r a r i l y small pos i t ive moment M along the middle p r i n c i p a l axis w i l l p u l l the spin vector back to the p o s i t i v e - h a l f and forces w to perform a per iod ic motion. In the perturbed case th i s i s no longer t r u e . As seen from Figure 3 « 1 2 , i f q Q i s s u f f i c i e n t l y large , m approaches the negative y-axis fo'r any a r b i t r a r i l y large pos i t i ve moment. It can assume per iod ic motions only for s u f f i c i e n t l y small values of q. Obviously i f the self-excitement M is negative, 3) w i l l approach the negative y-axis without any obs truc t ion . This fact is exhibi ted in Figure 3 . 1 3 -If a , = ~ a 2 ' then 35 w i l l move in the plane I I . Since [i =-2yacV, the equation determining k, becomes now (3-61) k , = jk~c~ a 2 [ 2 v ( a + c) - b] = -kg 1 b 1 * If V < ^ a Y c ' ) ^ ~2~' then sgn k , i s negative. The so lut ion of ( 3 . 5 9 ) can then be written as (3 .62) q 2 = C l e 2 _ _ _ | _ e 2 + _ J Equation ( 3 . 5 9 ) has a s ingular point on the<pg-axis at provided that m^  < 0. As before, we obtain for the equations of f i r s t approximation at the s ingular point the c h a r a c t e r i s t i c roots X = + yra^ • Thus the s ingular point i s a center. From 70 th i s fact and equation (3»62) i t follows that a l l other t r a j e c -tor i e s in Figure 3*14 are closed curves surrounding the s ingular point o If we compare th i s Figure with the corresponding Figure 3.5 for the unperturbed case, we see that per iod ic motions of 35 in plane II are preserved provided that i s s u f f i c i e n t l y smal l . Since in the unperturbed case the spin vector of a force -free asymmetric gyroscope tends to coincide with the pos i t ive y - a x i s , i t is reasonable to expect that for m^  > 0 and small values of |(x| th i s tendency in general w i l l be preserved. That th is i s ac tua l ly so i s shown by the phase-plane t r a j e c t o r i e s in Figure 3»15 for m^  > 0. However, because of the negative damping term -vq^ the magnitude of (?) cannot increase i n d e f i n i t e l y , but approaches the value , Let us assume now that is s u f f i c i e n t l y large so that V > 2(a+c) a n d s q n k l P ° s i t i v e . The so lut ion q^ is given by -2vq> - ? m (3-63) - C l e 2 + _ | _ e 2 + _ J ^ Equation (3.59) has a s ingular point on the 9 2 ~ a x i s at <P- = ln<i-) de m^  i f m^  > 0. The c h a r a c t e r i s t i c roots of the equations of f i r s t approximation at the above point are A. = + ^m^. Consequently, the s ingular point i s a saddle po in t . The t r a j e c t o r i e s in the phase plane are shown in Figure 3«l6. (See page 73«) The separatr ix subdivides the phase plane into four regions of motion. In the t h i r d and the fourth region, 3 tends asymp-t o t i c a l l y to the negative y-axis with modulus tending to i n f i n i t y . Figure 3 .15 7 2 In the remaining two regions, 35 tends to the pos i t i ve y-axis with a f i n i t e magnitude. When m^  < 0 , no s ingular point occurs in the phase plane. The constant non-posi t ive self-excitement coupled with the negative damping term completely overcomes any tendency of 35 to coincide with the p o s i t i v e y - a x i s , and 35, wherever i t s i n i t i a l p o s i t i o n is in plane I I , eventual ly coincides with the negative y - a x i s . Figure 3»17 i l l u s t r a t e s the phase-plane t raj ec tor ies . In the rare s i tua t ion when V = - / • — r , the constants k, and kg are zero. Equation (3«8) reduces to ,2,„ , J r f , v 2 = m where ||J-| = b | ^ a - c , or equivalent ly to the system ^ = m _ l „ l „ 2 (3-65) dt rao " I N q & = q dt q This system can be integrated to obtain the equation (3-66) q 2 - c 1 e " 2 l M ' l < P , c , constant def in ing the phase-plane t r a j e c t o r i e s . These are shown in Figure 3.18. As seen from Figure 3 « 1 8 ( c ) , when m^  > 0, 35 in general tends to the p o s i t i v e y-axis with a f i n i t e magnitude. The m (a+c) exception occurs when the i n i t i a l value q < 1 o — b s / a c 73 ra„ > 1 m„ = 1 0 < rn, < 1 Figure 3.16 Figure 3.18 74 the spin vector then tends to an axis l y i n g in plane II and perpendicular to the y - a x i s . For mQ < 0, i t tends to the above mentioned axis without any exception. In the c r i t i c a l case raQ = 0 (See Figure 3 . 18 (h) ) , we f i n d that w tends to zero along the pos i t i ve y -ax i s i f q Q i s p o s i t i v e , and to the above mentioned axis i f q i s negative. Furthermore, in th i s c r i t i c a l o case the whole <P-axis represents states of unstable equi l ibr ium motion. 3»5 General Remarks and Conclusions The above remarks conclude the discussion of the motion of a s e l f - e x c i t e d asymmetric r i g i d body subject to a time-dependent torque vector f ixed along the middle p r i n c i p a l axis 2 with component equal to M + U.Q|(3| . As in the preceding Sec t ion , we have shown the p o s s i b i l i t y of in tegrat ing Euler*s equations in terms of an a u x i l i a r y var iable Cp. If th is is followed by a quadrature and an invers ion , we are able to express p, q, and r as functions of the time t . Using the method of phase-plane ana lys i s , we were able to i l l u s t r a t e the d i f ferent types of motion which m could assume with respect to the body-f ixed t r i h e d r a l . In p a r t i c u l a r , we showed that the end-point of 35 always had to move e i ther on the surface of one of the conjugate hyperbol ic cy l inders or on one of the two planes of the separating polhodes of a force- free asymmetric gyroscope. Further i t was shown that m could assume stable p e r i o d i c motions and asymptotic motions of the f i r s t , second, and t h i r d k inds . 75 Having obtained p, q, and r as functions of (p, we can com-pute the Euler angles as functions of cp» F i n a l l y , we would l i k e to mention the p o s s i b i l i t y of a s l i g h t extension of the problems discussed in Sections 2 and 3 , Namely, i f we assume that the torque vector 18 is f ixed along the l arges t , or middle, or smallest p r i n c i p a l axis with components equal respect ive ly to f , ( q , r) + <?,(9# r)p , or 2 2 f 2 ( P » r ) * 9 2 ( P » r ) q # o r f-jCp* a) + <3J(P, , then we can obtain p, q and r as functions of t by means of quadratures and inversionso However, because of the general nature of the functions and g, ( i = 1, 2 , 3) we are not able to discuss in d e t a i l , as before, the q u a l i t a t i v e behavior of i3. There-fore , we s h a l l leave such problems with these remarks. 76 SECTION 4 PERIODIC SOLUTIONS 4 . 1 General Considerations In Sections 2 and 3 we establ i shed that under sui table conditions the spin vector m of an asymmetric gyroscope, subject to a time-dependent torque vector f ixed along one of the p r i n c i p a l axes, can assume per iod ic motions in a s u f f i -c i en t ly small neighborhood of a stable permanent r o t a t i o n , represented by a center in the corresponding phase plane . In th is Sect ion we are concerned with the* construction of these per iod ic so lu t ions , using the method of Poincare. For th i s purpose le t us consider f i r s t the d i f f e r e n t i a l equation to which a l l of our l a t e r equations can be reduced. Here k > 0, p\ ( i = 2, 3, 4), and are a r b i t r a r y constants. This equation belongs to a general class of. d i f f e r e n t i a l equations known as Lyapunov-systems [24]« In the (0, ~ ) phase plane of ( 4 « l ) , i t i s known [24] that in a s u f f i c i e n t l y small neighbor-hood of the o r i g i n the equation possesses per iod ic solut ions which are ana ly t i c functions of the i n t i a l value r) of 0, and reduce to the t r i v i a l so lut ion 0 = ~ = 0 for 0 = 0 . Further -on ' more, the per iod of each of these solutions i s also an analyt ic 21T function of r> , reducing to — when r~) = 0. We s h a l l proceed to the construct ion of these per iod i c solut ions for »9 £ 0. (4.1) 77 Since ( 4 « l ) i s autonomous, there is no loss of general i ty d0 in assuming that the i n i t i a l value of — is zero. Denote by T the per iod of the per iod i c solut ion 0 corresponding to the i n i t i a l value r j . We may write ( 4 . 2 ) T = |2T(1 + h,0 + h 2 r j 2 * h 3 r ) 3 + --.) where h, ( i = 1, 2, •••) are constant coe f f i c i ent s to be determined l a t e r . Fol lowing Malkin [ 2 4 ] * we s h a l l f i r s t make a change of the time scale by l e t t i n g ( 4 . 3 ) T = • h x r ) + h2rf + h y ) 3 + . . . ) then equation ( 4 « l ) assumes the form ( 4 . 4 ) * (1+h^+h . y 2 - . . ) 2 * ds f, v v 2 \2l^2 0* ?3 0 3 ^4 0 4\ /d0\2 - ( l - h l V h 2 0 • . . . ) (^2 2 31* ^2 41 ) + ^lldS) C l e a r l y , the per iod ic so lut ion 0 as a function of s i s ana ly t i c and of per iod 2 7 T . Thus we s h a l l seek the so lut ion 0 of (4«4) in the form ( 4 . 5 ) 0 = f) cos s + f ) 2 0 2 ( s ) +r) 30^ ( s ) + . . . where 0 . (s) are per iod ic functions of per iod 2w which have 1 d0 . (O) the i n i t i a l values 0 ^ 0 ) = ~ — - = 0, i = 2, 3, ••• • S u b s t i t u t i n g the expression ( 4 « 5 ) in ( 4 « 4 ) and equating the coe f f i c i en t s of l i k e powers of n , we obtain 78 ^ • • 2 - ^ 5 * 2 - ; + 2 V - - ^ 2 - ~ l 2 s For th i s equation to have a so lut ion of per iod 2ir, the c o e f f i -cient of cos s must be zero. We obtain then For the function 0^(s) we have cos s — + * \ - -r*- Icos 3 s 2 4 k 4 1 2 k 2 1 2 4 k 2 whence we f i n d as before 5 0 2 5P„ P\ M-2 h 2 K2 K3 1 2 l O T . 4 o , , 2 ^1 u , 2 6 ( 4 . 7 ) \ 5 7 6 k 4 576k- 1 1 9 2 k * * i 0 x ' P 2 P 2 2 2 + 2 1^ " 9 1^ ) c o s 2 8 3 6 k 4 18k 2 1 V 1 ^2 ^2 P 3 ^1 r - 5 H, - 5 + -57—) cos 3 s . 1 9 2 k 4 3 2 k * 1 1 9 2 k * h J 79 The constant h^ is obtained by equating to zero the coe f f i c i en t of cos s in the equation for 0 . ( s ) . This then k y i e l d s We can proceed in the same way as above to obtain higher order approximations,; however, the computations soon become long and complicated. Thus we s h a l l go no further , and consider that the required p e r i o d i c so lut ion 0 and i t s per iod T are expressed approximately by formulas ( 4 » 2 ) , ( 4 » 3 ) a n <3 (4-5) to (4.8) . Let us return now to the problem discussed in Section 2. 4.2 Per iodic Solut ions in the Case of the Torque Vector  Fixed Along the Largest P r i n c i p a l Axis It was shown that the components of w can be expressed in terms of the var iables <p or cp^  since the two are re la ted by the f i r s t equation of ( 2 . 1 l ) . Further , cp ^ s a t i s f i e s the d i f f e r e n t i a l equation (L 9) ^ 2 l d < ? 2 \ 2 (4.9) " a * - » ! • s in cp2 • * {-^ dT ' When jm^| < 1, i t was es tabl i shed that the s ingular points of ( 4 « 9 ) occurred on the cp^-axis, and some of these s ingular points were centers surrounded by closed t r a j e c t o r i e s which represent p e r i o d i c solut ions of (4 .9) . Our purpose here i s to construct these per iod ic so lu t ions , using the theory described in Sect ion 4»1» Putt ing these solut ions in the 80 expressions for p, q , and r , and replac ing T by tj2d, as defined in Section 2, we s h a l l obtain the components p, q , and r as p e r i o d i c functions of t . Such procedures avoid evaluat ing and i n v e r t i n g the i n t e g r a l in ( 2 . 1 7 ) . Denote by cp^c the value of cp 2 at a s ingular point of the center type, and let cp2 = ^ + 0. Then (4.9) becomes 2 ( 4 . 1 0 ) - d —| + / l - m 2 s in 0 = m 1 - cos 0 + ^("jf) Here we have used the fact that at the center cos cp 2 c i s equal j 2 to - , / l -ra^. If we replace s in 0 and cos 0 by 0 3 sin 0 = 0 - j j c o s 0 = 1 ~ fl + 41 then equation ( 4 . 1 0 ) assumes the form (4 .11 ) ^ • ^  , . B l f? . / ^ f £ - ^ g . ^ ( f f Comparing th i s equation with ( 4 » l ) , we can immediately write down, from formulas (4 .2) , (4*3) and ( 4 « 5 ) to (4 .8) , the per iod i c so lut ion 0 corresponding to the i n i t i a l value rj of 0 . Hence, the required per iod i c so lut ion Cp 2, occurr ing in a s u f f i c i e n t l y small neighborhood of the center, and i t s per iod T are determined by the fo l lowing formulas: 1 S = ( l - m 2 ) 4 ( l + h i r ) • h 2 ^ 2 + h ^ 3 + . . . ) ~ 1 T Cp 2 = (p 2 c + ")cos s + " ) 2 0 2 ( s ) + ^ 3 0 3 ( s ) + . . . 81 2 f 29m, 274m, , i o • 1 — 5 - • u. * ~ \ 5 7 6 ( l - m J ) 5 7 6 , / l - m J 2" **1T 192 * 216 -"1 1 C O S S 2 / m, m. 2 2 (4.12) + ( 9 t • 7==^ ^ - ^ J cos 2s 36(l-m 2) 187i-mf 1 2 2 ral ral 1 ^1 ^ - 192 + 24 ) c o s 3 s 192(l-m 2) 32,/l-m2 1 h l = ° v, 5 m l , 5 m l n h2 = ,„„ 2^ + n, *l + 16 + ~ 48(l-m 2) 24,/l-m2 5 m j 3 ° m l m l 2 2 „3 h„ TJ]~2 - i t _ ^ _ 2 n ^ ~ n /-—jl V-x ~ 9 1) 3 1 4 4 ( l - m 2 ) 3 / 2 144(l-m 2) 1 3 / l ^ f 1 2" ~ 12 48yi-m^ T " ( J ^ l / 4 ( 1 + h l H + ^ + h 3 1 3 + — )• In the above formulas i f we put u.^  = 0, we o b t a i n the p e r i o d i c s o l u t i o n c o r r e s p o n d i n g to the unperturbed case. F u r t h e r , i n case the torque i s f i x e d along the s m a l l e s t p r i n c i -p a l a x i s , the co r r e s p o n d i n g p e r i o d i c s o l u t i o n s can be obtained 82 by simply interchanging the appropriate q u a n t i t i e s , as mentioned in Section 2.60 A•3 P e r i o d i c Solut ions in the Case of the Torque Vector  Fixed Along the Middle P r i n c i p a l Axis Here we need to consider a number of subcases, because the behavior of 35 depends not only on the i n i t i a l pos i t i on of 35 r e l a t i v e to the two planes of the separating polhodes but also on the values of [ X , and m^  ( i = 1, 2, 3)« Although the d i f f e r e n t i a l equations s a t i s f i e d by the var iable cp 2 are s l i g h t l y d i f f erent in form, the steps taken to obtain the corresponding per iod ic solut ions are i d e n t i c a l with the ones in Sect ion 4 * 2 . Hence, in what follows we s h a l l simply sum-marize the resu l t s accordingly . Moreover, without any loss of genera l i ty , we s h a l l assume as in Section 3»4 that i s negative, i . e . [i^ = - V , V > 0 . 1. The i n i t i a l pos i t ion of 55 not in planes I and II ( i ) | ^ | < \i (see equation (3 .52)) I s - (1 • m 2 ) 4 ( l + h ^ • h 2 H 2 + h 3 r) 3 + . . . ) Cp2 = Cf>2c, • 'jcos s + >")202(s) + rj 3 0 (s) + • • -^ 0 = arc sinh m s 83 3 \l2(l+m 2 ) 2jl+*\ 3 f 2 9 m l + 2 ? 4 X v + 1 * V - - i - , cos s V 5 7 6 ( l + m 2 ) 5 7 ^ ^ ? 2 1 6 1 9 2 / ^ • m l u 2 2 . ;V - — V ) cos 2s V36(l+m 2 ) l & J l ^ ( ral m l v 2 1 \ (4.13) • 27 " 7 = = ^ V + 2 l + 192 C O S 3 8 \l92 ( l*m 2) 32,/l+mJ 2 4 1 9 2 / h, = 0 2 ^ = 48(l+m 2) + 2 4 ^ 1 ^ \ + 6 " 1 6 3 2 5 M 1 3 ° M 1 m l 2 2 3 h^ = —TTT? * 7 ^ V + j — 7 V + o v J 144 ( l +m*) J , /* 144/1 -ni* 371+m* y m . ± _ w 12 48,/l+m2 T - M , % i / 4 ( 1 + h i n + h 2 ^ 2 + h 3 n 3 + m " ] Putt ing V = 0 in the above expressions we obtain the corresponding per iod ic so lut ions in the unperturbed case. ( i i ) > H-c (See equation (3.56)) s = (m2 - 1 ) 1 / 4 ( 1 + h , r ) + h ^ 2 + h y ) 3 + - . . ) - 1 T , m 2 > 1 <P2 = c p 2 c + rjcos s + ^ 2 0 2 ( s ) + r j 3 0 3 ( s ) + • • • Cp 2 c - arc cosh c p 2 c > 0 84 "2 v , _ 1 9 / 2 . ~ 6 l c o s 2 s 0,(s) = - ( — — • -rr=v + I v 2 3 \l2 ( m ^ - l ) 2 y m J - l / 29nu 274m9 ift^ ? 2 m2 m2 2 2 ' A L - - v - -£v^ ) cos 2s 36 (m 2 - l ) l&7m |-l 9 i 2 m 192(m 2 - l ) " ^fifl ' 2 4 " 1 9 2 ^ m2 u v 2 + . v + — + — - c o s 3 s h 1 = 0 h - 5 m 2 + 5 m 2 v + vf 1_ 2 48 (m 2 - l ) 2 4 V m | - 7 6 1 6 h = 5 m 2 + 3° m2 , + m 2 v 2 + 2 v 3 _ m 2 _ v_ 3 1 4 4 ( m 2 - l ) 3 / 2 144(m 2 - l) 37m 2-1 9 4^ m * - l 1 2 h 0 n 3 + •••) ( m 2 ^ i / 4 ( 1 * h i n + h 2 ^ 2 + h 3 n ( i i i ) In the l i m i t i n g case = u^, per iod i c solut ions can be obtained from the fo l lowing case 2 since s a t i s f i e s an i d e n t i c a l equation. 2. The i n i t i a l pos i t i on of tt) in plane I (See page 68) 85 s = Jm^(l * hfl + h ^ 2 * h ^ 3 + . . . ) _ 1 T , m3 > 0 f2 = f 2 c • ^cos s + rf^2^a^ + r ) 3 ^ 3 ( s ) + ••• 9 2 c = In m3 ( 4 . 1 5 ) 03(.) " (- x i - i - | v 2 ) > ^ • | Z A v • ^ v 2 . cos . 1 • v 2 2 \ / 1 v v 2 \ 36 + 18 " 9 V ) C O S 2 s + ^96 " 32 + 2 4 y C ° s 3 s m v + 183v2 576 v 216 y - M + 96 • ' 32 + 2 1 _ 5 v h 2 * 24 * W * T T = — ( 1 • • h 2r) 2 + h r^> * •-.) Putt ing v = 0 we obtain the corresponding per iod ic solu-t ions in the unperturbed case. 3« The i n t i a l pos i t ion of «S in plane II (see page 69) ( i ) M < H c -s = y - m ^ U • h,r) + h2r) + h . ^ • . . . ) T, < 0 9 2 = cp 2 c ^ c o s s + rj 0 2 (s) • ry>0 (8) * ••• CP_ - In f- — • a ( . ) - i - j - i - § » ) ~ . . - r - i i * i « . (4.16) * ( 8 ) - ( - 1 - • - 2 A - • ^ v 2 ^ U - i o ; v y s ; ^ 1 2 2 3 / \576 576 v 216 y cos s '7 1 v 2 2\ / 1 v v 2 \ + ' 3 6 " 18 " 9 V C O S 2 s + 9 6 + j 2 + 2 4 C O S h l " 0 1 _j> h 2 " 2 4 " 2 4 V + 6 1 v 1 2 2 3 n 3 7 2 8 3 9 T = 27T -»3 ( l +11 ,0 + h 2r) 2 + h^rj3 + . . .) The corresponding per iod i c solut ions in the unperturbed case are obtained by put t ing V = 0, ( i i ) When | ^  | > [i, no per iod i c solut ions e x i s t . 87 SECTION 5 SELF-EXCITED SYMMETRIC RIGID BODY IN A VISCOUS MEDIUM 5.1 Equations of Motion We consider here the motion of a symmetric r i g i d body about a f ixed point in a viscous medium subject to an a r b i t r a r y time-dependent self-excitement• It w i l l be shown that the com-ponents of i« can be obtained by quadratures. In p a r t i c u l a r when the self-excitement i s constant and f ixed in d i rec t ion within the body, these components are express ible in terms of general ized sine and cosine i n t e g r a l s , whose values can be computed from i n f i n i t e ser ies developed in Section 5«4« Section 5 ends with a discuss ion of the asymptotic motions of «5 r e l a t i v e to the body-f ixed t r i h e d r a l . These motions are further compared with the corresponding ones when f r i c t i o n is neglected. We s h a l l see that the dif ferences be-tween the two are s i g n i f i c a n t . The resul ts of th i s discussion are summarized in three theorems. During the f i n a l stage of wr i t ing th is thes is we d i s -covered that Merkin [25] also discussed the problem considered in th i s S e c t i o n . The resul ts here were obtained independently, and inc lude , in addit ion to others , the resu l t s of [25]. In s e t t ing up the equations of motion of a body moving in a viscous medium, we need to consider the nature of the . .resist ing force generated by the movement of the body. For the case of a force- free symmetric r i g i d body ro ta t ing about 88 a f ixed po int , Kle in and Suramerfeld [16] assumed that the drag force was equivalent to a torque vector with two components, one along the symmetric axis and the other along the orthogonal projec t ion of m on the equator ia l plane. These components were assumed to be proport iona l to the components of (3 in these di rec'tions. In our study we s h a l l fol low th i s suggestion. Let the z-axis be the symmetric ax i s . Then we may denote the r e s i s t i n g components by -V-Q* a n d -u.^(p • q ) , where U . q and u.^  are con-stants which we w i l l assume p o s i t i v e , or i f f r i c t i o n is neglected zero. Further , suppose (5.1) a = M j C O * • M 2 (t)7 + M 3(t)lc where M^(t) ( i = 1, 2, 3) are continuous functions of t , is the s e l f - e x c i t e d moment act ing on the body. Then the equations of motion take the form Ap - (B - C)qr - M 1 ( t ) - p^p (5.2) Bq • (A - C)rp - M 2 ( t ) - u^q Cr = M (t) - P Q r T + ^ B-C A-C . *1 *1 u ^o A _ M l ( t ) Let t ing - j - = —g- ' h» X" " B~ " v# C~ " a n d f ^ U ) = ~ J ' M 2 ( t ) M (t) F 2 ( t ) = —g -, F^(t) = —* , we may write equations ( 5 « 2 ) in the form 89 p - hqr = F 1 ( t ) - V p (5 .3) Q hrp = P 2 ( t ) - vq r = P 3 ( t ) - [xr When the s e l f - e x c i t e d torque vector i s f ixed in d i r e c t i o n in the body as w e l l , we may put F 1 ( t ) = ^ F ( t ) = ^ ( t ) (5 .4) F 2 ( t ) « g^F(t) = A. 2 F(t) F 3 ( t ) = ^ F ( t ) = A. F( t ) Here a , , a^, and Q-^ denote the d i rec t ion cosines of the f ixed vector . In add i t ion , i f the act ing moment i s time-independent, F ( t ) i s a constant. Otherwise the torque vector w i l l move in the body; i t s pos i t ion at any time t i s given by e i ther ( 5 « l ) or ( 5 . 4 ) . Equations (5«3) together with the i n i t i a l angular ve loc i ty 3 Q ( p o , q Q , r Q ) at t = 0 determine uniquely the angular ve loc i ty of the body. We proceed then to the integrat ion of these equations. 5.2 The Angular Ve loc i ty of the Rigid Body The t h i r d equation of (5«3) can be integrated d i r e c t l y to give (5.5) r = r o e _ l i t + R(t) where 90 R ( t ) = r t e - ^ ( t " s ) F . ( s ) d s M u l t i p l y i n g t h e s e c o n d e q u a t i o n o f (5«3) by i and a d d i n g t h i s t o t h e f i r s t o n e , we o b t a i n (5.6) TT * 5ir = T T where IT = p • i q , TT = F, + i F 0 , and P = v + i h ( r e " ^ 1 • R) 1 <6 O a r e c o m p l e x - v a l u e d f u n c t i o n s o f t« T h i s l i n e a r e q u a t i o n h a s t h e g e n e r a l s o l u t i o n ~~S " 5 t ( 5 . 7 ) i r = 7 r e 1 • e 1 f e S ( s ^ T f ( s ) d s ° J 0 p t H e r e *5 •, - | 5 ( s ) d s a n d = P + i q • By s e p a r a t i n g t h e J. J Q o o o r e a l and i m a g i n a r y p a r t s o f (5*7) we s h a l l o b t a i n e x p l i c i t f o r m u l a s f o r p and q. We n o t e f i r s t : ~ "5, — v t * — v t 7 r Q e = e ( p o c o s ^ - * g o s i n 5 ) + i e ( - p ^ i n ^ s * q o c o s s ) ( 5 . 8 ) e'^.J te 5 l ( STT(s)ds - J V v ( t - s ) { F , (s)cos[5(s)-3(t )] t 5 , ( B ) | ( s ) d s * | 0 - F 2 ( s ) s i n [ 3 ( s ) - S ( t ) ] } d s • i j . t e ~ V ( t " s ) { F 1 ( s ) S i n C 3 ( s ) - 5 ( t ) j + F 2 ( s ) c o s [ ^ ( s ) - 5 ( t ) ] } d s w h e re t 0 ( 5 . 9 ) 5 ( t ) = f h [ r e^8 * R ( s ) ] d s 91 ~2 ~2 P o Let t ing 6 » J p + q and CT = arctan ~ " , we have o 6e V * s in ("5 • CT) 0 (5 . 10) - F 2 ( s ) s i n[3 ( s ) - 5 ( t ) ] } d s 6e V tcos ("5 + cr) • T * e - v ( t - s ) { F 1 ( s ) s i n [ 3 ( s ) - 3 ( t ) ] • F 2 ( s )cos [5 (s ) -5 ( t ) ] }ds ( i ) The Torque Vector Fixed in Direct ion Within the Body In the above ana lys i s , the s e l f - e x c i t e d torque vector i s not r e s t r i c t e d in any way. If we now assume that m i s f ixed i n d i r e c t i o n r e l a t i v e to the body so that (5.4) holds, then p o Se V t sin(s+cr) - 6 1 J o t e " v ( t " s ) F ( s ) s i n [ 5 ( s ) - S ( t ) - C T 1 ] d s (5.U) q = 6e V*cos(^+cr) • 6 j t e " V ( t " s ) F ( s ) c o s [ ^ ( s ) - 5 ( t ) - o 1 ] d s r ~ 2 — 2 ~ / ^ i \ where 6^ = / \^ * ^"2 a n d °1 ~ a r c t a n 1 ~ / " From (5.5) the expression for r becomes (5 .12) r -= r e"^ • \, f ( T ~ S >F (s )ds 92 ( i i ) The Resistance of the Medium Neg l ig ib l e When a r i g i d body moves in a medium which we may assume f r i c t i o n l e s s , then = = 0 , and from equation (5*9) . t (5.13) ~S(t) = f h[r + R(s)]ds Formulas (5.10) and (5.5) give . t p = 6 s i n ( 3 +a) • F , ( s ) cos[s ( s)-3 ( t ) ]ds - J* t F 2 ( s ) s i n [ ^ ( s ) - 3 ( t ) ] d s (5.14) . t q = 6 cos(-5+o) • I F n ( s ) s in[5(s ) -5( t )]ds J 0 x t F 2 ( s ) c o s [ ? ( s)-3 ( t ) ] d j + ' 0 t ds 0 In add i t ion , i f the torque vector 18 is f ixed in d i r e c t i o n within the body, the above formulas can be further s i m p l i f i e d . For now we have from (5.11) and (5.12) r 1 p = 6 sin("5+CT) " 6, J o F ( s ) s i n [ 3 ( s ) - 5 ( t ) - o 1 ] d s (5.15) q - 6 cos(s+cr) • 6, J F ( S ) C O B [ s ( s ) -5 ( t ) - a i ] d s r * f F(s) ds ° 3 Jn t 0 These formulas then provide the angular ve loc i ty components of a r i g i d body under the action of a time-dependent s e l f - e x c i t e d torque vector f ixed in d i r e c t i o n within the body. 93 In a l l of the cases considered above, the solut ions are given in terms of some i n t e g r a l s . The p o s s i b i l i t y of evaluating them by means of elementary or tabulated functions depends, of course, on the functions F^ , F^ , and F^, or on the funct ion F i f the s e l f - e x c i t e d torque vector i s f ixed in d i r e c t i o n . In the fo l lowing , we s h a l l consider th i s p o s s i b i l i t y in d e t a i l for the case of a constant self-excitement f ixed in d i r e c t i o n within the body. Afterward, we s h a l l discuss q u a l i t a t i v e l y the asymptotic motions of 3 subject to th is type of torque. 5.3 Time-Independent Torque Vector Fixed in Direct ion Within  the Body Denote by M the modulus of the torque vector. Let 2 2~ ral * m2° • r n e n f r o m formulas ( 5 « l l ) and (5.12) we have p = 6e" V t s i n (-s+a) - m 1 2 J e~ ~ s ^  s i n [ 5 ( s ) - 3 ( t )-CT^]ds (5.16) q = 6e~ V t cos(5+a)+m 1 2 J e " V ^ t _ s ^ c o s [5 ( s ) -5 ( t ) - a 1 ] d s In case \i. and v are n e g l i g i b l e , these expressions reduce to f t p «= 6 sin(^+a) - m 1 2 J s in["S(s) -^( t ) -a 1 ]ds (5.17) q = 6 c o s ( ^ a ) + m 1 2 c o « [ T ( s)-3 ( t ) - o 1 ] d s r = r o * m t 94 For the two in tegra l s appearing in (5.17), Bodewadt [2] showed that they general ly could be expressed in terms of the Fresnel i n t e g r a l s . For the corresponding in tegra l s appearing in (5.16) , we s h a l l l a t e r show that in the general case they can be expressed in terms of in tegra l s of the trigonometric type which includes the Fresne l i n t e g r a l s . We proceed now to discuss four cases, depending on the values of h, r and m„. ° 3 • Case 1. h ^ 0, r Q = 0, m^  = 0 From equations (5.9) and (5.16) we obtain / _ v t + m l fc l S i / m 2 \ ~vt ^ m2 (5.18) q - ^ q o - — j e • ~ r = 0 Case I I . h f 0, TQ f 0, m^  = 0 Under these condi t ions , we have (5.19) r = r o e _ l i t and -5(t) = h x ( l - e _ t i t ) hr where h.. = —-— . Without any loss of genera l i ty , we can assume i [i in the fo l lowing computations that h^ > 0 since the pos i t ive d i r e c t i o n of the z-axis may be selected so that hr > 0. o 9 5 For the i n t e g r a l appearing in the f i r s t equation of (5.16), put t ing 0 = h,e - o,# we may write (5.20) f t e - v ( t - s ) s i n [ 5 ( s ) - 3 ( t ) - a 1 ] d s J 0 -vt . ^ r t vs , -M-S , • e sin?) e cosh 1 e r ds - e V+"cos0 J* s inh,e ^ s ds 0 The l a t t e r two in tegra l s can be expressed in terms of the fo l lowing two funct ions: • ( \ P °° s in u . i (x . w) - - J U , w) - - J du ( 5 . 2 1 ) . , „ , - 0 0 cos u , c i ( x , j =  | • — - — du X u which we may consider respect ive ly as the general ized sine and cosine i n t e g r a l s . Note that for w = 1/2 they become the Fresnel i n t e g r a l s . We s h a l l allow w to take complex values . The ex i s -tence of these in tegra l s and the methods of evaluat ing them w i l l be considered in Section 5«4» In the meantime, we continue to show how they are used in expressing the components p and q . If we make a change of var iable by means of the equation —s u = h,e r the f i r s t and second integrals appearing on the r ight of (5*20) become respect ive ly J e V s cosh 1 e" M ' 8 ds = h 2 [ c i O^a^ - c i ( h^'^a,) ] ( 5 . 2 2 ) J" e V s s inh 1 e"" l i s ds = h 2 [ s i ( h ^ a ^ - e i (h^e"*^1,a,)] 96 v where a.. = 1 • 77, h_ = __1 . Thus we may write 1 * * \i (5.23) P = 6 e " V t s i n(-s* a ) - m 1 2 e ~ V t h 2 [ c i ( h 1 , a 1 ) - c i (h 1e"M' t ,a 1)]sin0 * m 1 2 e ~ V t h 2 ' - s i ( h 1 * a 1 ) ~ s i (h 1e" P' t #a 1)]cos0 For the component q we f i n d in a s i m i l a r way that (5.24) q = 6e~ V t cos (-s+a)+m 1 2e~V th 2[ci (h 1 , a 1 ) -c i (h 1 e~ | J ' t # a 1 ) ]cos0 + m 1 2 e " V t h 2 [ s i ( h 1 , a 1 ) - s i ( h e " ^ , )]sin0. Case I I I . h j<= 0, m^  ± 0 From formulas ( 5 « 9 ) and ( 5 « 1 6 ) we now have (5.25) r - ( r - ^2) e"^ + ^ -l i t ^ r ~ —/ mT. and ~S(t) = g(l-e~^ ) + g.^ where g = - ~ ^- , g1 = T ^ h . Suppose f i r s t g £ 0. Putt ing t = ge"1"1 - g x t - o x T r * v s - I A S . 1 = J e c o s g i s c o s g e T r 4 v s . -M-s, I 2 *s J e s in g^s cos ge r ds I 0 B f e V s c o s g,s s in ge ) i a ds 3 J 0 1 t 0 we observe that 1^  = J e V s s i n g^s s in ge ^ S d s 97 J t e " v ( t " s ) s i n [ T ( s ) - - s ( t ) - ( j ; L ] c l s - e " V t ( l 1 + I 4 ) s i n ^ + e " V t ( l 2 - I 3 ) c o s f J t e - v ( t - 8 ) C 0 S [ T ( s ) _ I ( t ) - 0 i ] d s = e " V t ( l 1 + I 4 ) c o s f - e " V t ( l 2 - I 3 ) s i n < f Consequently, for the components p, q, we have from (5.16) p = 6 e " V t s i n (^+o)-ni 1 2 e"" V t [ ( l 1 + I 4 ) s i n f + ( l 2 -I 3 )cos<f] (5.26) q = 6 e " V t c o s ('5+o-)+m12e"Vt[ (I , + 1^)008^- ( i g - I ^ J s i m f ] The in tegra l s I , to I. can be expressed in terms of the 1 4 general ized tr igonometric in tegra l s as fo l lows: M u l t i p l y i n g I 2 by i and adding th i s to I , , we obtain t * <T r t ( v + i g i ) s — -US . I , • i l 2 = J e cosEge ds where 6 = sgn g so that eg > 0. If we make the subs t i tu t ion u = ege~^ S, then (5 .27) I x • H 2 - M ' [c i (eg ,w 1 ) -c i (ege" t ) ' t ,w 1 ) ] V . i g l in which w. = 1 + 7- + —rr~, and the f i r s t factor on the r ight can be writ ten as _y i g l _y _y - lneg) + i s in f -T^lneg Put t ing V. tea)* / g l , P 98 c i ( e g # w x ) = U A + i V x c i ( e g e ' ^ w ^ = Ug + i V g and e q u a t i n g t h e r e a l and i m a g i n a r y p a r t s o f (5-27), we ha v e h - « 2< ui - V - « 3 ( v i • V <5-28> i 2 •= , 3 ( 0 , - u 2 ) • , a ( V l - v 2 ) I n a s i m i l a r way i f we l e t s i ( e g , W l ) = U 3 + i V 3 t h e n s i C e g e ' ^ w , ) = U 4 • i V ^ (5 29) ^ = , [ , * ' V V " « 3<W ]. i 4 - e [ ? 3 ( n 3 - u 4 ) * a 2 ( v 3 - v A ) ] T h u s , f o r m u l a s (5.26) t o g e t h e r w i t h (5.28) and (5.29) p r o v i d e t h e c o m p o n e n t s p and q when g f 0 . I n c a s e g = 0 , t h e s e c o m p o n e n t s can be e v a l u a t e d i n t e r m s o f e l e m e n t a r y f u n c t i o n s . F o r now we h a v e t / \ , - V t J e " V U " s ; s i n [ T ( s ) - T ( t ) - a 1 ] d s - " ^ s i n (o^+CTg) + — s i n ( g ^ t +0^0 2 t (t ) 1 e ~ V t J* e~- ~ S ' c o s [ T ( s ) - 5 ( t ) - o 1 ] d s = " Q ^ C O S (aj^+CTg) + " ^ 3— c°s (g,t + 0 1 + a g / 2 2~ g l where 6g = / V • g , , Qg = a r c t a n — . It f o l l o w s t h a t c — v t , r _ , ra12 - v t . ( f + + v , v m l ^ l m 2 p <= Oe s x n (g^+aj - ^ — e s i n (g,t *o1*02) *—2—2— (5 .30) V - v t m 12 - v t V m 2 " g l m l q = 6e c o s ( g . t + a ) + T — e c o s ( g . t • C T - . + O O • — ~ — 2 — 2 V+g* 99 Case IV. h = 0 This is the case of dynamic symmetry. D i r e c t l y from equations (5«3) we obtain ( , . 3 1 ) , . i . ( , o . i ) . - . t 5.4 The Integrals s i (x f f w) and c i ( x , w). Re w > 0 v In the problem under consideration Re w = 1 + — , v C 1^ * where the ra t io — = — 77— . If LI = u, or i f they do not O d i f f e r much, then for an elongated r i g i d body where A is much v greater than C the r a t i o 77 is less than 1; accordingly , the \r Re w i s between 1 and 2. For a f l a t - d i s k e d r i g i d body where now C i s greater than A, the Re w i s greater than 2. It i s known [18] that the in tegra l s s i ( x , w) and c i ( x , w) converge for Re w > 0. In the fo l lowing computations, l e t us assume that the V ra t io 7  i s not an integer so that we can write w = K + w* where K is a p o s i t i v e integer and w1 = a + i{3, 0 < a < 1. Then repeated in tegrat ion by parts y i e lds for K odd K+l (-1) 2 c i (x, w) =P (x) s in x-Q(x)cos X+ ( w '- i j,".'.w'ttlSi (x,wT )-A(w (5.32) K^l si(x,w)=P(x)cos x-Q(x)sin x+ (w'~'i'j'.".". w* ^- C i ( x ,w» )-B(w 100 and for K even K ci(x,w)=P(x)sin x-Q(x)cos X + j ^  <> - t [ Ci (x, w •) - B( w* ) ] (5.33) K si(x,w)=-P(x)cos x-Q(x)sinx+ ( ) ^ . . w t [ S i ( x , w ' ) - A ( w ' ) ] In these equation^ ^ 2 / . \n + l 2n-w P(X) = I (W:.r..(w-2n) i f K o d d (5.34) (5.35) and (5.36) (5.37) n = l K 2 V ( - l ) n * 1 x 2 n ° W  = L (w-l)...(w-2n) i f K e v e n n = l *>i 2 - (_n )n + l 2n-l-w Q ( X ) " I (w-~.j../(W-2n+l) i f K o d d n=l K 2 (, 1 )"+l x2n-l-w 1 (w-l)- . .(w-2n + l) i f K e v e n n = l S i ( x , w » ) = f -S-^P du 0 < Re w» < 1 J:p. u W x C i ( x , w » ) = [ du 0 < Re w' < 1 J 0 u W A(w») = f °° du 0 < Re w' < 1 J 0 u W B(w») « f -Sff-i du 0 < Re w* < 1 J 0 u W 101 It i s known [18] that for 0 < Re w' < 1 A(w») - TT 2 r ( w f ) s i n ( w » | ) B(w») = 1T Here P(wf) denotes the gamma function of w f . The integrals S i ( x , w f) and C i ( x , w f) appearing on the right of equations (5»32) and (5«33) have been studied for wf r ea l by Kreyszig [18, 19], and t h e i r values have been p a r t i a l l y tabulated . However, for wT complex no such tables have yet appeared [6 ]o Since these values are needed for the solutions of Case I I I , we s h a l l in the fo l lowing develop ser ies for the integra ls S i ( x , wT) and C i ( x , w f) and asymptotic series for the re lated in tegra l s s i ( x , w) and c i ( x , w). From these ser ies we may compute t h e i r values . In add i t ion , estimates of errors w i l l be provided when by necessity summations of the given series are c a r r i e d to only N terms. ( i ) T a y l o r t s Expansion for S i ( x . w) and C i ( x . w), 0 < Re w < 1 From the i n f i n i t e series n = l termwise in tegrat ion y i e lds 00 102 I f w * O • i £ , the ser ies on the r ight can be separated into rea l and imaginary p a r t s . For by putting x w = x a e * X 1 , where x^ = |3lnx, and n n = / ( 2 n - a ) 2 + ji1 A = arc tan J L n \2n-a we may write . x , " , ,xn+l 2n»a i ^ A n ~ X l ^ T s in u , V (-1J x e J0 — d U " I (8 .-1)1 ^ „ n = l or (5.38) Jo ^•-_I ( - ^ - l )m B ° " ( V ' l > " , . sn + 1 2n-a I (-&-iW'-»<v-i> n - l • i n = l For the second i n t e g r a l , the same procedures give „ x * / . xi i 2n*l-a . * / , \n 2n+l-a 0 ™ / , V A l l J a J. L U 4 - ' n ' n n=0 n=0 where now f l * « , / ( 2 n * l - a ) 2 + 0; A* = arctanfx .ft-n \2n + l - a ; The above procedures w i l l also y i e l d an estimate of the error when the summation of the ser ies (5*38) or (5«39) i s 103 c a r r i e d only to N terras. For i f we let N _ / .xn+1 2n-l-w s in u V" 1-1J u -w n U n = l in which R„ is the remainder after the Nth term in the sine N s er i e s , then terrawise in tegrat ion gives r A iS -J l d u = y - 1 ) ° u2n-l-w J n , w £ (2n-i;X J n 0 u ^. v ' ''O n = l „ x p — w • u R„ du. C l e a r l y , we have I < J Q X I U " W R N I D U From Lagrange*s remainder formula, we have where M is an upper bound on the absolute value of the 2Nth der iva t ive of s in u. Since th i s der ivat ive i s e i ther + sin u or - s in u, M can be taken as 1. Thus with u 2 N ' V ^ T2N7l 1 -w, -ti" |u I = u we may assert that 2N+1-U (5.40) I EN' $ ( 2 N)l ( 2 N U - a ) For the series (5 - 39) we f ind that 2N-tt (5.41) , | E N | < ( 2 N-i ) : (2N-a) 104 Because of the f a c t o r i a l terms in the denominators of the above two expressions, we can expect the series (5.38) and (5.39) to converge s u f f i c i e n t l y fast for small values of x. ( i i ) Asymptotic Ser ies for s i ( x . w) and c i ( x , w). Re w > 0 For large values of x, the values of the two integrals s i ( x , w), c i ( x , w) are best evaluated by means of asymptotic s e r i e s . For the i n t e g r a l s i ( x , w), repeated integrat ion by parts gives (5.42) J i i n u w du N - l 2 - • y n = l N - l 2 I n = 1 ( - l ) n w ( w + l ) « « » ( w + 2 n - l ) w+2n ( - l ) n * 1 w ( w + l ) « " (w+2n-2) w+2n-l cos x s in x N+l 2 + (-1) 2 w(w+l)-..(w+N+l) J — ^ d u . X u i f N odd. N-2 2 1_ + r ( - l ) n w ( w + l ) « " (w+2n-l) w _w+2n 2 I n = l n = l ( - l ) n * 1 w ( w H ) » « ( w t 2 n - 2 ) w+2n-l cos x sin x • (-1)2 w(w + l ) . - . ( w + N - l)J du X u i f N even. 1 0 5 If we let H ( x , w ) = i _ + y ( - l )%(w+l). . . (w+2n-l) then n = l 7 \ V ( - l ) n ' > ' 1 w ( w « - l ) " » (w+2n-2)  A X ' W ; ~ L w*2n-l n = l X C D ^ ^ ^ (5.43) ~—^— ^ H(w, W)COS x • G(x, w)sin x J x u The asymptotic nature of th i s expansion can be seen from the considerat ion of the remainder term. If R„ denotes the remainder after the sum of the f i r s t N + 1 terms, then from (5.42) |R NI - |w(w + l ) . (w+N)[°° ~ j ^ d u l i f N + 1 odd . X u = | w ( w + l ) . « . ( w + N ) j 8 ^ " H " du| i f N • 1 even - x u w In e i ther case, we have | R NI < |w(w*l ) - - - (wN)|J~ - ^ T l . 0 0 - (N+a) = | w ( w + l ) • » • ( w * N ) | X M + a  From th© r e l a t i o n (5.44) Rw • 1) - wP(w) for the gamma funct ion , we may write 106 (5.45) P ( W * N ^ I ) r(w) t - (H*a) N+a N, which i m p l i e s t h a t |x R N| - 0 as x -• <» f o r every N. We may note t h a t the v a l u e s of the gamma f u n c t i o n f o r complex arguments w have been e x t e n s i v e l y t a b u l a t e d [27] so t h a t e q u a t i o n (5.45) i s a c o n v e n i e n t e s t i m a t e of the remainders The s e r i e s H(x, w) and G(x, w) can be s e p a r a t e d i n t o r e a l and i m a g i n a r y p a r t s as f o l l o w s : We observe f i r s t t h a t w i t h r e l a t i o n ( 5 * 4 4 ) , we may w r i t e A r e ) * 2 " - 1 * " P u t t i n g -n_(w) = |r(w) | A(w) = arg p(w) w a 1 x = x e where x, = 01nx, we have then OD H(x,w) * £ ( " l ) n ^ i ^ 2 n ) c o » ^ ( w ^ n ^ A ( w ) - x l 3 n=0 "^(w)3 1 n«=0 (5.46) + i J ^ 1 ^ ^ 2 n ) , i n [ A ( w . 2 n ) - A ( w ) - x 1 ] G(x,w) = Y ( " l ) n ^ I i " " l ) c o 8 [ A ( w ^ 2 n - l ) - A ( w ) - x ] ^, l~L(w)x n = l v ' 107 For the second i n t e g r a l c i ( x , w), an asymptotic series can be obtained in the same way. It i s found that - 0 0 (5.47) I C ° * "du ^ -H(x,w)sin x • G(x,w)cos x J w d X u with an estimate of the remainder given again by (5.45). F i n a l l y i f we let H(x, w) = + i H 2 G(x, w) = + i G 2 then si(x,w) „ -H 1 cos x - G . s in x - i(H_cos x + G 0 s i n x (5.48) 1 1 2 . . . 2 ci(x,w) ^, H^sin x - G^cos x + i ( H 2 s i n x - G 2cos x 5«5 The Asymptotic Motion of the Spin Vector When a r i g i d body rotates in a viscous medium, i t s angular ve loc i ty w i l l in general be l imi t ed by f r i c t i o n . This statement i s substantiated by the fo l lowing boundness Theo rem: The angular ve loc i ty of _a s e l f - e x c i t e d symmetric  r i g i d body ro ta t ing about a f i xed po in t in _a viscous medium i s  bounded, i f the modulus o f the s e l f - e x c i t e d to rque vector i s  bounded. The truth of th i s theorem follows eas i l y from the expressions for p, q and r . For from equation (5.10), we have 108 | p | < | 6 e - V t S i n ( r * a ) | • I f V V ( t " B } F . ( . ) c o « [ < f ( . )-5(t ) ] d s J o -+ l j V V ( t " " } F 2 ( , ) 8 i n ^ ( 8 ) - 5 ( t ) ] d s . , - V t _ - V t M t P* V s . _ - v t M I p 1 V s , < Oe • e M* I e d s • e MI e d » where | F 1 ( B ) | < M l , | ( s ) | < M« . Integrat ing the expressions on the r i g h t , we f i n d that M* M' | p | < 6e + — (1-e ) + — (1-e ) which in turn implies that M» M» | P ! < 6 + * ~{p for a l l t > 0 S i m i l a r l y , we obtain M» M» | q | < 6 + ~ + ~ for a l l t > 0. From equation ( 5 * 5 ) , we have r < r e -M-t, + , - l i t J 0 If F 3 ( t ) < M», then M» r l < Ir I + -4 for a l l t > 0. It i s e a s i l y seen that this theorem need not hold i f f r i c t i o n i s neglected. Moreover, even though the speed of rotat ion is bounded, the spin vector may s t i l l wander in space without approaching any l i m i t . Nevertheless, i f the s e l f -excitement i s time-independent and f ixed in d i r e c t i o n r e l a t i v e to the body, we s h a l l show in the fo l lowing that 3? approaches asymptot ical ly a constant vector in the body-fixed t r i h e d r a l . 109 The r i g i d body i t s e l f then assumes asymptotical ly a uniform rotat ion about an axis f ixed in the body as well as in space. We s h a l l consider th i s in four cases as in Section 5»3» and in doing so we s h a l l also compare these motions with the corresponding ones in which f r i c t i o n is neglected. There are e s sent ia l d i f f erences . Case I . h * 0, r Q = 0, m^  = 0 Since r = 0, the motion of the end-point of 55 takes place wholly in the equator ia l plane. The system p - hqr = m^  - Vp (5.49) Q • hpr = m2 - vq f = - Lir with the assumption nu = 0 possesses a unique s ingular point / m l m2 \ ( ~ , — , Ol in the pq plane. According to (5.18) every t ra jec tory in th i s plane is r e c t i l i n e a r and approaches the s ingular point as i l l u s t r a t e d in Figure 5 . 1 . This point represents a state of permanent rotat ion of the body, and this rotat ion i s s tab le , as we s h a l l see in the discussion of Case I I . When f r i c t i o n is n e g l i g i b l e , r remains i d e n t i c a l l y zero, and the end-point of 35, as before, always l i e s in the pq plane. But the phase p ic ture i s e s s e n t i a l l y d i f f e r e n t . For now the system has no s ingular point on the pq plane, unless m^  = = 0. From (5.17) we have P = P Q + m 2t q - q o + m 2t 110 implying that in the pq plane the t r a j e c t o r i e s are p a r a l l e l l ines tending to i n f i n i t y . These are shown in Figure 5«2. This example v e r i f i e s our previous statement that the angular v e l o c i t y need not be bounded i f f r i c t i o n is neglected, even though the modulus of the s e l f - e x c i t e d torque vector is bounded. Case I I . h * 0, Xq t 0, m^  = 0 Turning t o the second case, we consider f i r s t the f r i c t i o n l e B S s i tua t ion s i n c e t h e motion i s t h e n simpler t o descr ibe . With L> = V = 0 and IQ $ 0, equation (5.19) gives r = r Q . Thus, t h e e n d - p o i n t of 3), i n i t i a l l y l y i n g i n t h e plane z = r Q , w i l l remain forever in t h i s plane. The f o l l o w i n g system, obtained from (5-49) by p u t t i n g V, |X, a n d m^  zero. p - hqr = m, (5.50) q + hrp = m 2 r =0 has in the phase space (p, q, r) a set of s ingular points cons i s t ing of the space curve defined as the in tersec t ion of the two hyperbol ic cy l inders " m l qr = - r -m2 pr = T — Furthermore, since q 112 (5.51) £ m _2 q m. th is curve l i e s wholly on the plane defined by (5•51)• The nature of th i s curve in the plane can be seen i f we rotate - m 2 the pq plane about the r -ax is by an angle 0 = arc tan « In the rotated coordinate system, the equation of the curve is given by m l q 1 cos Q - p' sin 0 = (5.52) m 2 ' q* s in Q + p* cos 9 = h"rT Here p '» q*, and r T denote the new coordinates . Equation (5«52) i s equivalent to q » r r = k' = constant p' = 0 and hence th i s curve is a hyperbola as i l l u s t r a t e d in Figure 5»3» in which the p*-axis i s perpendicular to the paper. This curve intersects the plane r = r Q (^  0) at the / m2 -m1 \ point I - , , r I • The t r a j e c t o r i e s in th i s plane form a o o I continuum of concentric c i r c l e s with the center at the s ingular po in t . For from equations (5.17)/ we have p = 6 s in(hr Q t+a) + ^~{]m,s in(hr o t ) + m 2[ 1-cos (hr Q t ) ]] o , A , = arctan 113 m 2 m q o + h ~ o then m 2 P - s i n ( h r Q t + A ) o In the same way, we have m l q + T = A . c o s i h r t + A , ) hr 1 o 1 o These las t two equations define parametr ica l ly a c i r c l e of radius Consequently, in the plane z = r , the end-point of t» moves in a c i r c l e . The phase t r a j e c t o r i e s in the plane r = TQ are shown in Figure 5»U* Moreover, since r i s constant, these c i r c l e s represent a continuous family of per iod ic motions about a stable permanent r o t a t i o n . The per iod of each of these motions is c l e a r l y T = 2 i r |hr | ' o ' which varies inverse ly with r for a given mass d i s t r i b u t i o n , o represented by the constant h . From the resul ts of Cases I and I I , we have the fo l lowing Theorem : For _a s e l f - e x c i t e d symmetri c r i g i d body rotat ing  about _a f ixed point in _a f r i c t i o n l e s s medium, i f the t ime-independent torque vector is f ixed in the equator ia l plane, the  mot ion of the end-point of t» t akes p i ace who 11 y in the i n i t i a l  plane z = r Q . In the phase space (p, q, r ) , the t r a j e c t o r i e s in  the plane r = r Q form a continuum o f concent r i c c i rc les with 114 Figure 5«4 115 nig -m^ canter at the s ingular point ( — , — , r ) , representing _a o o state of st able permanent rot at i o n . If r Q = 0, the center being considered at i n f i n i t y , these c i r c l e s degenerate into p a r a l l e l l i n e s . In the case of a force- free symmetric r i g i d body, we know that when r £ 0 the end-point of t» also moves in a plane o perpendicular to the z -ax i s , and i t s locus in th i s plane form concentric c i r c l e s with center on the z-axiso Thus we observe that geometrical ly one effect of a time-independent s e l f -excitement with f ixed d i r e c t i o n r e l a t i v e to the body is to s h i f t the center of these c i r c l e s to a new locat ion in the plane of motion. Consider now the s i tua t ion when f r i c t i o n is not n e g l i g i b l e . The motion of the end-point of m becomes more complicated. For now, since r = r Q 9 the motion no longer takes place in a plane perpendicular to the z - a x i s . Instead, the end-point approaches asymptot ical ly the xy plane . As mentioned in Case I , the system (5*49) with m^  = 0 has m l m2 only one s ingular point TJ ly one s ingular point ( ~ » ~ , o) • On the basis of the lemma in page 1 17 *nd formulas (5«16), we observe that as t -» a> the components p, q, and r approach respect ive ly the values m l m2 — , — , and 0. Thus, every t ra jec tory in the phase space approaches the s ingular po in t . Case I I I . h * 0, m^  * 0 If f r i c t i o n i s neglected the system (5.49) has no s ingular 116 p o i n t . Every t r a j e c t o r y in the phase space (p, q, r) tends to i n f i n i t y . The components of m are now given by formulas (5.17). As the two in tegra l s on the r ight of (5.17) can be expressed in terms of the Fresnel integrals and the tr igonometr ic - funct ions , and since these two kinds of functions are bounded for t > 0, the magnitudes of p and q are thus bounded a l so . On the other hand, | r | increases i n d e f i n i t e l y when t -» » , and as a result the vector aj approaches asymptotical ly the pos i t ive or negative z - a x i s , according as the a lgebraic sign of m^  i s pos i t ive or negative. However, i f the body is further acted upon by a torque due to f r i c t i o n , the system (5«A9) w i l l then possess a unique s ingular point which can be obtained by so lv ing th© system of equations Vp - hqr - m^  = 0 (5-53) v<? + hpr - m2 = 0 LIT - m^  = 0 The so lut ion of (5-53) i s m l V * m 2 g l p = 5 5— , where g, = —~-* m 2 V - m l g l r V-The fo l lowing lemma w i l l show that a l l other t r a j e c t o r i e s tend asymptot ical ly to th is p o i n t . 117 Lemma: For "s(t) = g^t - g e _ p , t + g, V > °# v > °» »s t - » ( i ) J . - f t 9 ~ V ( t ~ " ^ » i n [ ^ ( » ) - S ( t ) - C l ] d B - - — ^ ( V . i l l C T n + g i C O . C T i ) ( i i ) J 9 = f e~V^ t~*^cos[3(s)-3(t)-a,]ds - g 1 gCvcoaoT-qi^iaai,) v +gj_ Proof ; Integrat ion by parts y i e l d s sincr. -v t J x TT^ - • V " 8 i n ^ ( ° > - 5 ( t > - a i ^ - * g . " v t jV ( ^ v ) 8 cos [5( . ) -5 ( t ) -a 1 ]d ( • j V v ( t " " ) c o . [ 3 ( . ) - 5 ( t ) - a 13d. The t h i r d term on the r ight approaches zero i s t •* » . For we h ave | e - V t J t . - ( * - v ) 8 c o . ^ ( . ) - 5 ( t ) - a 1 ] d » | < •- V tJ t.- (» i- v ) Bd. 0 - V t Since the r ight hand side tends to zero when t -» <», the le f t hand side does a l so . Consequently, we may write g. - s i n a i (5.54) Jx * -^J2 " —v + e l ' w h e r a e i •* 0 " 1 "* The same procedures y i e l d 118 coso"i ~ J 2 7 ^ - • £ 7 ~ c o s ^ ( ° ) - ^ t ) - a 1 ] g • J J g e - V t j V ^ ' K i n ^ s ) - ^ ) - * ^ 1 J t e - v ( t - 8 ) 8 i n [ ^ ( B ) _ ^ ( t ) _ C T i - l d s V "0 Because the second and t h i r d terras tend to zero as t -* • , we may wri te g . cosCT, (5.55) J 2 - - — g — • * 2 ' w h e r e e 2 "* 0 a s 1 "* Solv ing for and from (5-54), (5.55), we obtain J"l = "~2^ ~2 ^ V s i n C T l + g i C O S a l ^ + E3* where -» 0 as t -» «o r and J o = ' o' 1 o ( V C O S C T I - g . s ing . ) + e , where £, -» 0 as t -* » . 2 V 2 + g 2 4 Thus from formulas (5•16) and the above lemma, we have as t -* <*> P = vm 1+g 1m 2 2 2 v +g^  V m 2 - g l m l q = 2 2 v^+gj Since a l l other t r a j e c t o r i e s tend to the s ingular po int , the state of permanent rotat ion represented by th i s point i s c l e a r l y s tab le . The'above p ic ture of the phase t r a j e c t o r i e s can be compared with the streamlines of a three-dimensional steady incompressible flow a l l tending to a p o i n t - s i n k . 119 In the case when the i n i t i a l pos i t ion of the end-point of m3 t» l i e s in the plane z = ~ , then equation (5o25) shows that i t w i l l remain forever in tRis p lane . In the phase space (p, q , r) th i s means that every t r a j e c t o r y or ig inated at any point in the m plane r » 77*% which contains the s ingular point of the system, w i l l remain forever in th i s p lane . Furthermore, m u l t i p l y i n g the f i r s t equation of (5•30) by i and adding th i s to the second one, we have V l ° 2 - g l m l \ ^ . ( v i y g l m 2 \ . . -vt 1 < V ° ) m 1 2 -vt i(»rt*0l*(l2) = e -Vt oe - e 62 The term in the bracket can be writ ten as i < T l m12 i(o1+a2) / m l g l ' v r a 2 \ / V r a l + g l r a 2 6e - j — . e = I 2—2" • q I * i ( - 2 2 * Pol 2 \ v^+g* 7 \ vA*q1 7 If we let 6^  and o*^  denote respec t ive ly the modulus and the argument of th i s vector , then / / v m i ^ i m 2 \ -vt i C ^ i * * ^ ) (q - 2 2 / \ p ~ 2 2 / = 6 3 e e \ V * * g * / \ V^+g, / •> m This equation shows that the t r a j e c t o r i e s i n the plane r = - ~ s p i r a l toward the s ingular p o i n t , as i l l u s t r a t e d in Figure 5»5» Case IV. h = 0 In the case of dynamic symmetry and n e g l i g i b l e f r i c t i o n , we have from (5»3) 120 P P a g, > o q Figure 5.5 P = PQ • m xt q = r = r + m„t o These expressions show that the angular v e l o c i t y increases i n d e f i n i t e l y at a uniform r a t e . Again th i s increase w i l l be r e s t r i c t e d when f r i c t i o n i s considered. For equations ( 5 « 3 l ) show that a l l t r a j e c t o r i e s in the phase space are r e c t i l i n e a r , and tend asymptot ical ly to The q u a l i t a t i v e resu l t s obtained in Sect ion 5*5 can be conveniently summarized into the fol lowing Theorem: For a s e l f - e x c i t e d symmetrie r i g i d body rot at ing  about _a f ixed point in a viscous medium, the mot ion of the  body approaches asymptot i c a l l y a uniform rot at ion about an  axi s f ixed in the body as we .11 as in space. The angular  ve loc i ty components along the x, y, z axes of the body-fixed the only s ingu lar point 121 V m l + g l m 2 V m 2 ~ g l m l M 3 t r i h e d r a l are respect jve ly 5 5—* o o—' TL ' V^+g~ M * If f r i ct ion i s neglected, the angular ve loc i ty in general  increases i n d e f i n i t e l y , except in the case when the torque vector  l i e s in the equatori a l p l ane with the i n i t i a l pos it'ion of ts\ out- side t h i s plane and the body is not dynamically symmetric. In such _a case, e i ther a> assumes _a st able per iod ic motion with per iod 27T equal to T = hr o or the body assumes ja st able pe rmanent rot at ion m 2 ~ m l with , ^ , r Q _as the angular v e l o c i t y components in the 0 0 d i r e c t i o n of the x, y, z axes r e s p e c t i v e l y . 122 BIBLIOGRAPHY 1. Andronov, A . , C h a i k i n , S o , Theory of o s c i l l a t i o n s , Moscow, 1937, Engl i sh t rans la t ion by Lefschetz , S . , Princeton Univers i ty Press, Pr inceton , 1949. 2. Bodewadt, U. T „ , Der syrametrische K r e i s e l bei z e i t f e s t er Drehkraft , Math. Z . , 55 (1952), pp. 310-320. 3. Bogoliubov, N. N., Mitropolsky, Y. A . , Asymptotic methods in the theory of non- l inear o s c i l l a t i o n s , t rans la ted from Russian, Hindustan Publ i sh ing Corporat ion, De lh i , India , 196.1. 4. Byrd, P. F . , Friedman, M. R . , Handbook of e l l i p t i c in tegra l s for engineers and p h y s i c i s t s , S p r i n g e r - V e r l a g , B e r l i n , 1954. 5. Coddington, E . A . , Levinson, N . , Theory of ordinary d i f f e r e n t i a l equations, McGraw-Hill Book Company, I n c . / New York, 1955. 6. F l e t c h e r , A . , M i l l e r , I . C. P . , Rosenhead, L . , Comrie, L . J . , An index of mathematical tab les , v o l . I and I I , 2nd e d . , Addison-Wesley Publ i shing Company, Inc . , Reading, Massachusetts, 1962. 7« Go lds te in , H . , C l a s s i c a l mechanics, Addison-Wesley Publ i sh ing Company, I n c . , Reading, Massachusetts, 1959. 8. Golubev, V. V . , Lectures on integrat ion of the equations of motion of a r i g i d body about a f ixed po in t , t rans la ted from Russian, Off ice of Technical Serv ices , Washington 25, D. C. 9. Grammel, R . , Der K r e i s e l , F r e i d r i g Vieweg und Sonne, Brunswick, Germany, 1920. 10.,, , Die stationSren Bewegungen des selbsterregten Kre i se l s und ihre S t a b i l i t S t , Ingen• -Arch . , 21 (1953), pp. 149-156. 11. , Der se lbsterregte unsymmetrische K r e i s e l , I n g e n . - A r c h . , 22 (1954), PP. 73"79. 12 , Ein Problem des sebsterregten unsymmetrischen K r e i s e l s , Annal i di Matematica, 50-51 (i960), pp. 187-196. 123 13o , Drehzah1abhangige Selbsterregung des unsymmetrischen K r e i s e l s , Ingen . -Arch . , 29 ( i 9 6 0 ) , pp. 1 5 3 - 1 5 9 -1 4 . Hurewicz, W., Lectures on ordinary d i f f e r e n t i a l equations. The Technology Press of the Massachusetts Ins t i tu te of Technology, Cambridge, 1 9 5 8 . 1 5 . Kamke, E . , D i f ferent ia lg le ichungen: Losungsmethoden und Losungen, L e i p z i g , Akademische Ver lagsgese l l schaf t , 1 9 4 3 « »t 16. K l e i n , F . , Sommerfeld, A . , Uber die Theorie des K r e i s e l s , 4 v o l s . , L e i p z i g , 1 8 9 7 - 1 9 1 0 . 1 7 . Koshlyakov, V. N . , On certa in p a r t i c u l a r cases of in tegrat ion of E u l e r ' s equations, connected with the motion of a gyroscope in a r e s i s t i n g medium (Russian), P r i k l . Mat. Mekh., 17 ( 1 9 5 3 ) , PP- 137 -148 . tt 18. Kreysz ig , E . , Uber den allgemeinen Integrals inus S i (z, a ) . Acta Math . , 8 4 - 8 5 ( 1 9 5 0 - 5 1 ) , pp. 117 -181. 1 9 . , Der allgemeine Integralkosinus C i ( z , a ) . Acta Math. , 8 9 - 9 0 ( 1 9 5 3 ) , PP- 1 0 7 - 1 3 1 -2 0 . Leimanis, E . , Dynamical systems. Lecture notes ( 1 9 6 0 - 6 1 ) , Un ivers i ty of B r i t i s h Columbia, Vancouver, Canada. 2 1 . , Non-l inear d i f f e r e n t i a l equations. Lecture notes ( 1 9 6 0 - 6 1 ) , Univers i ty of B r i t i s h Columbia, Vancouver, Canada. 2 2 . , Minorsky, N . , Dynamics and non- l inear mechanics, John Wiley and Sons, I n c . , New York, 1 9 5 8 . 2 3 . MacMil lan, W. D . , Dynamics of r i g i d bodies, Dover P u b l i c a t i o n s , Inc . , New York, 1 9 3 6 . 2 4 . Malkin , I . G . , Some problems in the theory of non- l inear o s c i l l a t i o n s , A E C - t r - 3 7 6 6 , Book 1 and 2 , Office of Technica l Serv ices , Washington 2 5 , D. C . 2 5 . Merkin, D. R . , On the theory of s e l f - e x c i t i n g gyroscopes, t rans la t ed from Russian, Off ice of Technica l Serv ices , Washington 2 5 , D. C. 26. Minorsky, N . , Non-l inear o s c i l l a t i o n s , D. Van Nostrand Company, I n c . , Pr inceton , New Jersey, 1 9 6 2 . 124 2 7 . National Bureau of Standards, Applied math, series 34, Table of the gamma function for complex arguments, U. S. Department of Commerce, Washington 25, D. C , 1954» 28. Stoker, J . J . , Non- l inear v i b r a t i o n s , Interscience Publ i shers , New York, 1950. 29. Whittaker, E . T . , A n a l y t i c a l dynamics of p a r t i c l e s and r i g i d bodies, 4th e d . , Cambridge Univers i ty Press , New York, 1937-

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