Alexander Mielke and Philip Holmes- Spatially Complex Equilibria of Buckled Rods

8/3/2019 Alexander Mielke and Philip Holmes- Spatially Complex Equilibria of Buckled Rods

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Spa tially Co m plex Equilibria o f Buckled Ro dsALEXANDER MIELKE & PHILIP HOLMES

1 . Introduction

In this paper we study the spatial aspects of equilibrium states exhibited byinfinitely or arbitrarily long rods, buckled by loads applied at their ends. Ourmethods exploit the Hamiltonian structure of the equilibrium equations and useregular perturbation methods based on completely integrable cases (MELNIKOVtheory). We obtain a qualitative description of classes of solutions close to suchlimiting cases, which correspond to geometrical symmetries and the vanishingof certain stress components. Our results imply that there exist spatially irregularor chaotic equilibrium states for rods under the appropriate load conditions.KIRCHHOFF [1859] was apparently the first to remark the analogy betweenthe equilibrium equations of a rod loaded at its end and the equations o f motionof a heavy rigid body pivoted at a fixed point. In this analogy the arclength alongthe axis of the rod plays the r61e of a time-like coordinate. LARMOR [1884] sub-sequently extended the analogy to rods with initial curvature and twist; cf. Love[1927, w167 The discussions in KmCHHOFF and LOVE assume linearconstitutive relations and the equations thus obtained are precisely analogousto the rigid body equations. However, as we show in the next section, the analogyextends to general nonlinear hyperelastic materials. The (non-canonical) Hamil-tonian structure of the rigid body equations is preserved, while the quadraticHamiltonian is replaced by a general function. This structure, and the existenceof certain integrals, derive from underlying symmetries and group structures inthe problem. While the derivation of the rod equations is relatively well known(cf. ANTMAN & KENNEY [198l], ANTMAN [1984], MIELKE[1987]), the Hamiltonianstructure is not normally emphasized and so we outline it in Section 2. Here wemean the Hamiltonian structure of the static problem with the arclength as time-like variable, in contrast to the dynamic problem which is a partial differentialequation with time and arclength as independent variables. See KRISHNAPRASAD,MARSDEN, SI io [1986] for the Hamiltonian structure in that case.While there is an elegant non-canonical formulation for the three degree offreedom rigid body equations (cf. HOLMES & MARSDEN [1983]) we find it more


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3 2 0 A. MIELKE ~; P. HOLMESc o n v e n i e n t h e r e t o w o r k w i th a r e d u c e d t w o d e g r ee o f f r e e d o m s y st e m i n c a n o n i c a lc o o r d i n a t e s . I n g e n e r al t h i s s y s t em i s e x p e c t e d t o b e n o n i n t e g r a b l e ; i n f a c t o u rm a i n r e s u lt s p r o v e t h i s t o b e t r u e n e a r c e r t a i n l i m i t in g c a se s. H o w e v e r , i n t h e t w oc a s es o f z e r o r e s u l t a n t f o r c e a n d o f c ir c u l a r s y m m e t r y , a n a d d i t i o n a l i n t e g r a lc a n b e f o u n d a n d t h e e q u a t i o n s s o l v e d c o m p l e te l y . T h e f o r m e r e a se c o r r e s p o n d st o t h e a b s e n c e o f g r a v i t a ti o n a l f o r c e s ( m o m e n t s ) a n d t h e l a t t e r to t h e w e l l k n o w n' L a g r a n g e ' t o p ( G oL D S T EI N [ 19 80 , C h a p t e r 5 ]) . I n b o t h c a s e s s m o o t h m a n i f o l d so f h e t e r o c l i n i c o r h o m o c l i n i c o r b i t s e x i st , a n d u s i n g t h e p e r t u r b a t i v e t e c h n i q u e so f M E L N IK O V [ 1 96 3 ] i n t h e H a m i l t o n i a n c o n t e x t o f H OL M ES & M A RS DE N [ 1 98 2 ,1 9 83 ], w e a r e a b l e to p r o v e t h a t t h e s e m a n i f o l d s b r e a k t o g i v e t r a n s v e r s e h o m o -c l i n i c o r b i t s i n t h e p r e s e n c e o f s m a l l r e s u l t a n t f o r c e s o r a s y m m e t r i e s . T h e n , b ya r g u m e n ts f a m i l i a r in d y n a m ic a l s y s te m s (SMALE [1 9 6 3 , 1 9 6 7 ] , GUCKENHEIMER &H O LM E S [ 1 98 8 , C h . 4 - 5 ] ) , i t f o l l o w s t h a t s p a t i a l l y c h a o t i c e q u i l i b r i u m s t a t e so c c u r .

T h i s p a p e r is o r g a n i z e d a s f o l lo w s . I n S e c t i o n 2 w e o u t l i n e th e d e r i v a t i o n o ft h e e q u i l i b r i u m e q u a t i o rt s a n d d i s c us s t h e H a m i l t o n i a n s t r u c t u re . I n S e c t i o n 3 w ep e r f o r m o u r f ir st r e d u c ti o n , o b t a i n i n g a c a n o n i c a l t w o d e g r ee o f f r e e d o m H a m i l -t o n i a n s y s te m . T h e m a i n r e s ul t s a r e g i v e n i n S e c t i o n 4 , t o g e t h e r w i t h a d i s c u s s io no f t h e i r p h y s i c a l i m p l i c a t io n s , i n c l u d i n g a r o u g h d e s c r i p t i o n o f t h e s p a t ia l s h a p e se x h i b i t e d b y r o d s i n s u c h ' c h a o t i c ' s t a te s . T h e r e m a i n d e r o f th e p a p e r is d e v o t e dt o p r o o f s o f t h e t w o m a i n t h e o r e m s o f S e c t io n 4 . S e c t i o n 5 c o n t a i n s a b r i e f o u t -l i n e o f t h e s e c o n d r e d u c t i o n a n d a p p l i c a t i o n o f M E L NIK OV 'S m e t h o d t o t h e r e s u l t-i n g p e r i o d i c a l l y p e r t u r b e d s in g le d e g r e e o f f r e e d o m s y s t e m . A d e t a i le d t r e a t m e n to f t h i s m a t e r i a l i s c o n t a i n e d i n H O LM E S & M A R SD EN [ 1 98 2 , 1 98 3] . T h e t w o m a i nt h e o r e m s a r e t h e n p r o v e d i n S e c t io n 6 a n d 7 b y c o m p u t a t i o n o f M e l n i k o v fu n c t i o n sf o r a p p r o p r i a t e l i m i t i n g c a s e s . C o m p u t a t i o n a l d e t a i l s a r e r e l e g a t e d t o t h e A p p e n -d i x . I n o r d e r t o m a k e e x p l i ci t c o m p u t a t i o n s w e r e s tr i c t o u rs e lv e s t o s t re s se s w h i c ha r e s u f fi c ie n t ly s m a l l i n m a g n i t u d e , s o t h a t l i n e a r e l as t ic i ty d o m i n a t e s . T h e g e o m e -t r i c n o n l i n e a r i t i e s a r e , h o w e v e r , u n r e s t r i c t e d .

2 . Th e Ro d Eq u a t io n s in Ha mi l to n ia n Fo r mT h e m o d e l o f a r o d t r e a t ed i n t h is p a p e r t a k es t h e f o l l o w i n g f o r m ( e f . A N T -

MAN [19 84 ], KRISHNAPRASAD,MARSDEN, & S IMO [1 9 8 6 ]) . W e c o n s id e r a p r i s m a t i c ,e l a st i c b o d y w i t h r e f e r e n c e c o n f i g u r a t i o n Q = R 3 o r .Q = I 3,w h e r e t E R o r t E I ~ R d e n ot e s t h e a x ia l v a ri ab l e a n d x = ( x l , x 2 ) E Z ' ,t h e c r o s s - s e c t io n Z ' b e i n g a b o u n d e d d o m a i n i n R 2. W e a s s u m e t h e d e f o r m a t i o n so f t h e r o d h a v e t h e a p p r o x i m a t e f o r m ~0: . q , f 2 ~ R 3, w i t h

~ ( t , x ) : r (t ) + R ( t) (X o ) . (2 .1 )H e r e t h e r ( t ) E R a d e n o t e s t h e p o s i t i o n o f t h e d e f o r m e d a x is a n d R ( t ) E S O ( 3 )s p ec i fi es th e p o s i t i o n o f t h e r i g i d l y t r a n s f o r m e d c r o s s- s e ct i o n . S e e F i g u r e 1. N o t eth a t d3 - -- -R ( t ) e 3 , t h e l o c a l c o o r d i n a t e a x is p e r p e n d i c u l a r t o t h e t r a n s f o r m e dc r o s s - s e c t i o n , n e e d n o t b e t a n g e n t t o r'(t) . T h u s w e a l l o w f o r s h e a r d e f o r m a t i o n s .


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Com pl ex Equ i li b r ia o f Buck l ed Rod s

el

F i g . 1. T he r od m ode l .

321

H o w e v e r , t h e l o ca l c o o r d i n a t e s y s te m { d r , d 2 , d 3 ), d i ~ R ( t ) e~ , i s s t il l o r t ho go na ls o t h a t d 3 ( t ) = d ~ ( t ) a s i n A N T M A N'S f o r m u l a t i o n .

T h e r e a r e d i f f er i n g c o n v e n t i o n s f o r d e f i n i ti o n o f s tr a in s . W e a d o p t t h e f o l l o w -i n g . W e s e t

v = R r r " - e 3 , ~ 2 = R r R ' ( 2 . 2 a , b )s o t h a t Q = - O r E s o( 3) , a n d f i n a l l y d e f i n e u b y

u = g 2 a ( 2 . 2 c )

~ W O Wm - - n - - ( 2 .3 )~ u ' a V '

t h e c o m p o n e n t s o f m b e i n g b e n d i n g a n d t o r s io n m o m e n t s -a nd o f n , s h e a r a n de x t e n s i o n f o r c e s .T h e e q u i l i b r i u m e q u a t i o n s f o r a r o d l o a d e d o n l y a t i ts e n d s a r e e a s il y e x p r e s s e di n t e rm s o f t h e f o r c e a n d m o m e n t v e c t o r s F , M i n s p a ti a l c o o r d i n a t e s ( e l , e 2, e 3) .T h e y s im p l y ex p r e ss t h e fa c t t h a t F a n d M a r e c o n s t a n t s . T h u s , v i a ( 2. 1 ), i n b o d y

f o r a r b i t r a r y v e c t o r s a E R 3. S i n c e u a n d I 2 a r e r e l a t e d b y ( 2 . 2 c ) , w e s o m e t i m e sw r i t e g J ( u ) s u b s e q u e n t l y . T h u s u = u ( t ) , v ~ v ( t ) a r e b o t h 3 - v e c t o r s ; t h e y a r et h e s t ra i n s in b o d y c o o r d i n a t e s . S p e c i fi c a ll y , u~ a n d u 2 re p r e s e n t b e n d i n g i n t h ed 2 , d 3 a n d d 1 , d 3 p l a n e s r e s p e c t i v e l y a n d u 3 i s t o r s i o n ( = b e n d i n g i n d x , d 2 ) ;v x a n d v 2 a r e s h e a r s i n t h e d l , d 2 d i r e c t i o n s a n d v 9 i s e x t e n s i o n i n t h e d 3 d i r e c t i o n ;e l . A N T M A N (~ K E N N E Y [1981, w 2] .T h e e l a st ic p r o p e r t i e s o f th e r o d a r e d e s c r i b e d b y a s t r a i n e n e r g y f u n c t i o nIV (U , v ) : R 6 - - ~ R , w h i c h i s a s s u m e d t o h a v e a n o n d e g e n e r a t e m i n i m u m a t( u , v ) ~ ( 0, 0 ), c o r r e s p o n d i n g t o t h e u n d e f o r m e d s t a te r ( t ) = t e a , R ( t ) -~ LT h e s t re s se s in b o d y c o o r d i n a t e s a r e t h e n g i v e n b y


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3 2 2 A. MIELKE d~ P. HOLMESc o o r d i n a t e s , w e h a v e

F : R n - --- co ns t , ( 2 . 4 a )( 2 . 4 b )

w h e r ej = _ j r : ( g 2 ( m ) Q(On ) )\ g 2 ( n ) , Q ( a ) b = a x b . (2 .11)

A l t e r n a t i v e l y ( 2 .1 0 ) c a n b e w r i t t e n i n t e r m s o f t h e L i e - P o i s s o n b r a c k e t d e f i ne d i nH O LM E S ~ ; M A R SD E N [ 1 98 3 , e q n s . ( 3 . 1 7 - 1 8 ) ] f o r t h e h e a v y r i g i d b o d y . I n t h a tc a s e t h e H a m i l t o n i a n i s s i m p l y H = 89 m 2 ]\ I i ] - k M g l n a, b u t w e n o t e t h a t t h e s y m -p l e ct ic s t r u c tu r e p e r m i t s f o r m u l a t i o n o f e q u a ti o n s f o r g e n er a l H a m i l t o n i a n f u n c -t i o n s .

R e g a r d l e s s o f t h e p r e c is e f o r m o f H , t h e g r o u p s t r u c t u r e im p l i e s t h a t t h e t w ofu nc t io ns I~ = In 12 and I2 - -- - m 9 n a r e c o n s t a n t s o f m o t i o n f o r ( 2 .1 0 ), a l o n gw i t h t h e H a m i l t o n i a n H i ts e lf , a s o n e c a n r e a d i l y v e ri fy . T h e s e a r e t h e o n l y q u a n -t it ie s d e r i v a b l e f r o m F = c o n s t a n d M = c o n s t , t h a t a r e i n v a r i a n t u n d e r m o -t io n s in S O ( 3 ) x R 3 ; i . e . , i n v a r i a n t u n d e r c h a n g e s i n R a n d r . ( N o t e , F - + R F ,

( r a n ) ' : J ( m , n ) V H ( m , n ) , ( 2 . 1 o )

M = R m + r = c o n s t .D i f f e r e n t i a t i n g ( 2 . 4 a ) , w e h a v e R ' n + R n " = 0 o r

n " = - - R T R ' n ~ - Q n = n u . (2 .5 )D i f f e r e n t i a t i n g ( 2 . 4 b ) a n d a p p l y i n g R r f r o m t h e l e ft y i e ld s R r R ' m - ? m " +R T ( r ' x R n + r x ( R n ) ' ) = 0 o r , i n v i e w o f ( 2 . 4 a ) ,

R r R " m + m " + R r r " x n = 0 . ( 2 . 6 )U s i n g ( 2 . 2 ) e q u a t i o n ( 2 . 6 ) b e c o m e s

m ' : - Q r n + n X ( e 3 - [- v ) : m x u n x ( e 3 -Jr- v ) . (2 .7 )E q u a t i o n s ( 2 .5 ) a n d ( 2 . 7 ) i n v o l v e t h e s t re s s e s a n d t h e s t r a i n s , b u t , a s W h a s a

n o n d e g e n e r a t e m i n i m u m a t u : v : 0 , i t i s l o c a l ly c o n v e x a n d h e n c e t h e r e l a -t i o n s ( 2 . 3 ) a r e l o c a l l y i n v e r t i b l e . W e m a y t h e r e f o r e w r i t e t h e i n v e r s e s

u = u ( m , n ) , v = v ( m , n ) ( 2 . 8 )a n d t h e r e f u r t h e r m o r e e x i s t s a r e a l v a l u e d f u n c t i o n H ( m , n ): R 6 --->R s u c h t h a t

8 H c ~ Hu = 8 m ' e 3 + v : 6 --~ -. ( 2 . 9 )H a n d W a re r e la t e d v ia a L e g e n d r e t r a n s f o r m ; i n th e d y n a m i c a l a n a l o g y , W p la y st h e r6 1e o f a L a g r a n g i a n a n d H o f t h e H a m i l t o n i a n ( c J ] A R N O L D [ 1 9 7 8 ] ).

W e c a n n o w w r i t e th e e q u i l i b r i u m e q u a t i o n s ( 2. 5) , (2 .7 ) en t i re l y i n t e r m s o ft h e s t r es s e s a s a n o n - c a n o n i c a l H a m i l t o n i a n s y s t e m


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Com pl ex Equ i l i b r i a o f Buck l ed Rod s 323M ---> R M + r R F ) . I n t h e r i g i d b o d y a n a l o g y I t c o r r e s p o n d s t o c o n s e r v a t i o no f m a g n i t u d e o f t h e g r a v it y v e c t o r a n d 12 t o c o n s e r v a t io n o f th e c o m p o n e n to f a n g u l a r m o m e n t u m i n t h e d i r e c ti o n o f t h e g r a v it y v ec t o r.R e m a r k . I n t h e a b o v e d i s c u s s i o n i t i s n o t c l e a r h o w t h i s s p e c i a l H a m i l t o n i a ns t r u c t u r e i n ( 2 .1 1 ) ar is e s. H e r e w e p o i n t o u t b r ie f ly h o w i t c a n b e d e d u c e d u s i n gt h e m e t h o d s f o r H a m i l t o n i a n s y s t e m s o n L i e g r o u p s a s d i s c u s s e d i n A B R A H A M~; MARSDEN [1978, C h. 4] .

I n o u r c a s e t h e L i e g r o u p is t h e E u c l i d e a n g r o u p S O ( 3 ) 3, i .e . t h e g r o u po f r ig i d t r a n s f o r m a t i o n s w i t h m u l t i p l ic a t i o n ( R 2 , r e ) ( R ~ , r l ) = ( R 2 R 1 , R2r 1 -~ 1"2)T h e L a g r a n g i a n L is g iv e n b y L ( R , r , R ' , r ' ) = W ( R r R , R r r ' - e3 ) . S i nce Wi s i n v a r i a n t u n d e r t h e a c t i o n o f G , i t is m o r e c o n v e n i e n t t o u s e t h e " b o d y c o o r d i -n a t e s " ( b a s i s ( d ~ , d 2 , d a ) ) t h a n t h e " s p a t i a l c o o r d i n a t e s " ( w i t h b a s i s ( e l , e 2 , c a ) ) ,i .e. u a n d v + e a a r e t h e st r a in s i n b o d y c o o r d i n a t e s . N o w L c a n b e w r i t t e n a sa f un c t i o n L : G x g ~ R ; ( R , r , l " 2 ( u ) , v + e a ) - + W ( u , v ) , w h er e g = T e Gi s t h e L i e a l g e b r a o f G w i t h L i e b r a c k e t [ (f 2 1 , v 2 ) , ( f 2 2 , v 2 )] = ( f~ 1 12 2 - Q 2 Q 1 ,~ 9 ~ v 2 - 122vl) .S i m i l a rl y t h e c o r r e s p o n d i n g H a m i l t o n i a n H is d e f i n ed o n G ( g * = d u a lo f g ) r a th e r t h a n o n t h e c o t a n g e n t b u n d l e T * G . T h e v a r i a b l e s i n g * , b e i n g c o n -j u g a t e t o t h e v a r i a b le s i n G , a re e x a c t l y t h e s t re s se s m a n d n i n b o d y c o o r d i n a t e s .O f co u r s e H = H ( m , n ) i s g i ven a s i n ( 2 .9 ) .

T h e p r i c e f o r w o r k i n g i n b o d y c o o r d i n a t e s m u s t b e p a i d w h e n t h e H a m i l-t o n i a n s t r u c t u r e o f G i s c a l c u l a t e d , s i n c e " i n e r t i a l e f f e c t s " a p p e a r . U s i n gT h e o r e m 4 .4 .1 o f A BR AH AM & M A RS DE N [ 1 97 8 ], w e o b t a i n t h e s y m p l e c t i c f o r m

t r t !t O ( R , r , r ( m ) , n ) ( ( R 1 , r l , ~ r ~ l , h i ) , ( R 2 , r 2 , ~ r ' ~ 2 , / ' / 2 ) )= R r R ' I : t ~ - R r R ~ : ~ 1 + R ~ r ' l " n 2 - R r r ~ 9 n ~

T I T r _ _+ r (m) : (R T R 'IR rR ~ - - R rR ~R T R ~) + n . (R R I R r 2 R r R ~ R r r ~ )w h e r e ~t~ 1 " ~r~2 : t r (~ r~T~2) an d / ' ( m ) b : m b .

T h e a s s o c i a t e d v e c t o r f i e l d X n i s d e f i n e d b y, , 0 H ~ H , O H O H( D ( R , r , O , U ) ( X H , (R2, r2, Q 2, n2)) : ~- ~: R2 -]- ~ r 2 - [ - ~ ' F " ~ -~ 2 - ~ ~ ' - n " n 2 "

A d i r e c t c a l c u l a t i o n r e s u l t s in~ H O H 8 H7 " , r l pR ' = R - ~ , = R -~ n ~ -~ n ,

(2.12)r , = r ~ O H ( ~ H ~ )o-T ~-fi r + 89 n | ~-d ~n | n ,w h i c h a r e e x a c t l y t h e e q u a t i o n s e s ta b l is h e d a b o v e , w h e n / ' a n d m a r e i d e n ti f ie d .I n a d d i t i o n , w e r e m a r k t h a t t h e i n v a r ia n t s F = R n and M - - - - - R m + r a r e e x ac t ly t h e m o m e n t u m m a p p i n g s o b t a i n e d f r o m t h e i n v a ri a n c e o f L u n d e r t h ea c t i o n o f G ( A B R A H A M & M A R SD E N [ 1 97 8 , T h e o r e m 2 . 12 ] ).


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3 2 4 A . MIELKE & P. HOLMESI n t h e a n a l y s i s b e l o w w e f i n d i t c o n v e n i e n t t o f i x I t = a 2 , 12 ---- a b a n d w o r k

w i t h t h e r e d u c e d H a m i l t o n i a n s y s t e m r e st r i ct e d t o t h is ( f a m i l y o f ) f o u r m a n i f o l d ( s ) .I n S e c t i o n 3 w e s h o w t h a t a s y m p l e c t i c s t r u c t u r e e x i s ts s u c h t h a t t h i s r e s t r ic t i o n i sa c a n o n i c a l t w o d e g r e e o f f r e e d o m s y s te m . H o w e v e r , t h e l i m i t in g c a s es w h i c h w es t u d y b e l o w a r e b e s t d e s c r ib e d i n i ti a ll y i n t e r m s o f t h e n o n - c a n o n i c a l c o o r d i n a t e s( m , n ) .

T h e s e t n = 0 i s c l e a r l y a n i n v a r i a n t m a n i f o l d f o r ( 2 . 1 0 ); re s t r i c t e d t o t h i sm a n i f o l d t h e e q u a t i o n b e c o m e s

m ' : m u ( m , 0 ) , ( 2 . 1 3 )w h i c h h a s t h e a d d i t i o n a l i n t e g r a l I a - --- I m 12. F r o m M m L K E [ 19 8 7 ] w e k n o w t h a t ,w i t h o u t l o s s o f g e n e r a li t y , w e c a n t a k e u t o h a v e t h e f o r m

u ( m , O ) : ( ~ x lm ~ , or or -~- o ( I m ]2 ) (2 .14 )w h e r e 0 < ~ 1 ~ ~ 2 a n d o~3 > 8 9 q - ~ 2 ) . H e n c e t h e r e a r e t w o g e n e r i c c a s e s :~ t < ~ 2 < c~3 a n d ~ 1 < or < or W e w i l l a l s o c o n s i d e r t h e s p e c i a l c a s ec~t : ~ 2 < ~ a , w h i c h o c c u r s i n t h e c a s e o f c e r t a i n c r o s s - s e c t i o n a l s y m m e t r i e s .T h e p h a s e p o r t r a i t s o n s p h e r e s / 3 : c o n s t a n t , s u f fi c ie n t ly s m a l l , a r e t h e n q u a l i -t a t i v e l y i d e n t i c a l t o t h o s e f o r t h e E u l e r e q u a t i o n s f o r t h e g r a v i t y f r e e r i g i d b o d y( G o L D S TE I N [ 1 9 8 0 , C h . 5 ] ). F i g u r e 2 s h o w s t h e c a s e ~ 1 < o~2 < ~ 3 . N o t e t h eh e t e r o c l i n i c o r b i t s c o n n e c t i n g t h e h y p e r b o l i c f i xe d p o i n t s a t m 2 : 4 - 12 , m ~ :m a : 0 . S o l u t i o n s c a n b e w r i t t e n d o w n e x p l ic i t ly i n t e r m s o f e l li p ti c f u n c t i o n s( W H I T T A gE R [1 9 37 , w 6 9 ] ) a n d t h e h e t e r o c l i n i c o r b i t s a r e h y p e r b o l i c f u n c t i o n s( c f . H O L M E S M A R S D E N [ 1 9 8 3 ] ) .

m 2m l

Fi g . 2 . H e t e r o c l i n i c c y c l e s f o r t h e p u r e b e n d i n g c a s e .

T h i s l i m i t c o r r e s p o n d s t o a r o d i n p u r e b e n d i n g a n d t o r s i o n w i t h z e r o sh e a r a n de x t e n si o n . T h e f i x ed p o i n t s m 2 = 4 - 1 2 c o r r e s p o n d t o c i r c u la r l y c o i le d r o d s :o n e o f t h e c l a s si c p l a n a r e q u i l i b r i a f o u n d b y E U L ER [1 7 4 4] .

I n t h e c a s e 0r < ~ 3 < ~ 2 t h e p h a s e p o r t r a i t s a r e s i m i l a r t o F i g u r e 2 , b u t t h eh e t e r o c l i n i c o r b i t s n o w c o n n e c t t h e f ix e d p o i n t s m ---- ( 0, 0 , 4 - m 3 ) . W h e n o q ----c~2 < ~ a t h e p o r t r a i t d e g e n e r a t e s t o a f a m i l y o f c i rc l es p a r a ll e l t o t h e e q u a t o r i a lc i r c l e m 2 q - m 2 ---- c o n s t . , m 3 ---- 0 a n d m 3 i s a n a d d i t i o n a l c o n s t a n t o f m o t i o n( s e e b e l o w ) .


7/30

Complex Equ i l i br i a o f B uck led Rod s 325T h e e x i s t e n c e o f t h e s e i n t e g r a b l e l im i t s s u g g e s ts t h a t w e c o n s i d e r a r o d w i t hs m a l l s h e a r a n d e x t e n s i o n . L e t t i n g

n---- eft , IK] = 1 (2.15)a n d d r o p p i n g t h e b a r s , w e p u t ( 2 . 1 1 ) i n t o t h e f o r m

m " = m u ( m , , n ) + , n (e 3 + ~ ( 0 ) , (2.16)n ' = n u ( m , e n ) ,A n a n a l y si s o f t h i s s y s t e m l ie s b e h i n d o u r f ir s t m a i n r e s u lt , T h e o r e m 4 .1 .A s e c o n d i m p o r t a n t c a s e i s p r o v i d e d b y r o d s w h o s e c r o s s - s e c t i o n s a r e s y m -m e t r i c w i t h re s p e c t t o t h e d i h e d r a l g r o u p D N w i t h N ~ 3 . ( T h u s , Z ' i s i n v a r i a n tu n d e r r o t a t i o n b y 2 ~r / N ) . I n t h a t c a s e t h e t w o c o n s t a n t s 0q a n d ~ 2 , w h i c h a p p e a ra t l e a d i n g o r d e r i n t h e f u n c t i o n u ( m , 0 ) o f ( 2 .1 4 ) a r e n e c e s s a r i l y e q u a l . I f w e t r u n -c a t e t h e H a m i l t o n i a n a t q u a d r a t i c t e r m s , t h e n e q u a t i o n ( 2.1 6 ) a d m i t s t h e a d d i t io n -a l i n te g r a l m 3 , w h i c h r e s u lt s f r o m t h e r o t a t i o n a l s y m m e t r y o f t h e r o d a t t h a t o r -d e r . T h i s c o r r e s p o n d s t o t h e i n t e g ra b l e L a g r a n g e to p , m a b e i n g t h e c o m p o n e n to f a n g u l a r m o m e n t u m a l o n g th e a x i s o f s y m m e t r y , a n d 1 /0 r 1 /~ 2 b e i n g t h e ( e q u a l )m o m e n t s o f i n e rt ia . O u r s e c o n d m a i n r e s ul t, T h e o r e m 4. 2, c o n c e r n s t h i s c a se .H e r e t h e p e r t u r b a t i o n p a r a m e t e r i s t h e s c a l i n g p a r a m e t e r t~, w i t h I n ] = a z a n dm - - - - 6 r h . A s ~ - + 0 w e a p p r o a c h t h e r o t a ti o n a l ly s y m m e t r ic H a m i l t o n i a n

^ 2-- ~ (~ im i) + n3 w ith ~1 = o~2( = o 0. T i le t e r m o f l o w e s t o r d e r t h a t b r e a k sc i r c u l a r s y m m e t r y w h i l e r e s p e c ti n g D n - s y m m e t r y i s t h e n ~ n - 2 R e ( rh l + ir~ 2)N.

T h e f a c t t h a t H c o n t a i n s t e r m s o f h i g h e r o r d e r i n t h e n o n l i n e a r e la s t ic c a s em a k e s t h e p r e s e n t a n a l y si s m o r e s u b t le t h a n t h a t o f t h e d y n a m i c a l a n a l o g u e c a r r i e do u t b y H OL ME S & M A R SD E N; f o r e x a m p l e d e l i c a t e s c a l in g a r g u m e n t s , c a r r i e d o u ti n S e c t i o n 3 , a r e n e c e s s a r y t o p r o v e T h e o r e m s 4 .1 a n d 4 . 2 .S o l u t i o n o f t h e H a m i l t o n i a n s y s t e m (2 .1 0 ) d oe s n o t s p e c i fy th e s p a t ia l s t a t e o ft h e r o d . E q u i p p e d w i t h t h e s t r es s e s r e ( t ) a n d n ( t ) a s f u n c t i o n s o f a r c l e n g t h t E B( re s p e c ti v e ly I ) w e m u s t t h e n i n t e g r a te t h e r e l a t i o n s ( 2 .2 a , b )

r" = R ( v ( m , n ) + e 3 ) ( 2 . t 7 a )R ' = R g 2 ( u ( m , n ) ) , ( 2 . 1 7 b )

O H ~ Hu s i n g t h e f u n c t i o n s u = ~ a n d v - - ~ n e 3 f r o m (2 .9 ). T h i s y i e l d s r = r ( t ) ,R --- R ( t ) a n d h e n c e , v i a ( 2. 1) , t h e d e f o r m a t i o n ~ ( t , x ) .

3. Firs t Reduct ion: A Symplect ie Transformat ion

T h e H a m i l t o n i a n s y s t e m ( 2 .1 0 ) h a s a d e g e n e r a t e s y m p l e c t ic s t r u c t u r e a n d i s,i n f a c t, a p a r a m e t e r i z e d f a m i l y o f t w o d e g r e e o f f r e e d o m s y s te m s . T o r e v e a l th i se x p l i c i t l y a n d m a k e e l e m e n t a r y c a l c u l a t i o n s p o s s i b l e , w e w i s h t o r e d u c e v i a t h em o m e n t u m m a p p i n g w h i c h d e f i n e s t h e i n v a r i a n t s /1 = i n [z a n d 12 ---- m . n( A B R A H A M d ~ ; M A R S D E N [1978, w 4.3]) .


8/30

3 2 6 A . M I E L K E P . H O L M E SL e t th e r e d u c e d p h a s e s p a c e b e d e n o t e d

Pa,b = ( ( f l t l , l r /) ~ ] ~ 6 [ I n 1 2 = a 2 , m . n = a b } ( 3 . 0 )f o r r e a l c o n s t a n t s a ~ 0 , b E R . W e a re e s p e c ia l ly c o n c e r n e d w i t h l i m i t i n g c a s e si n w h i c h a d d i t i o n a l i n t e g r a ls t m l a n d m a a r is e . T h i s s u g g e s t s t h a t w e p i c k a c o o r -d i n a t e s y s t e m i n P,~,bw h i c h h a s 1m l a n d m 3 a s tw o o f t h e n e w v a r ia b le s . W e d e f in et h e s e v a r i a b l e s ( r, s , a , ~ ) b y

i C O S rm : s i n a ,

S(3 .1 a )

( i o , ( s c o s a ' ~ { - s i n a ~a b coS a~si?a] - - r ~ s s i ) f f / c o s o + a--~-~ ~ C o a ] s i n 0 , ( 3 . 1 b )n - -~ --~ a r 2 r

w h e r e r : I / r z - s 2 a n d r : 1 /rS - b 2 . T h u s r n3 : s , I m l = r a n d i t c a nb e v e r i f i e d t h a t I n [2 : a z a n d m - n : a b , s o t h a t t h e p o i n t ( m , n ) : ( m ( r , s , a ) ,n ( r , s , a , Q )) l i e s i n P a , b a s r e q u i r e d .

T o o b t a i n t h e s y m p l e c t i c f o r m ~ o n t h e r e d u c e d s p a c e P a,b i n (r , s , a , ~ ) c o o r -d i n a t e s , w e n e e d th e t r a n s f o r m a t i o n m a t r i x

~ ( r , s , a , ~ )G - - 8 ( m , n ) ' ( 3 .2 )w h e r e e l e m e n t s a r e c o m p u t e d b y i n v e r s i o n o f e q u a t i o n s (3 .1 a , b ). T h e v a r i a b l e sr , s a n d cr a r e e a s i l y o b t a i n e d i n t e r m s o f m , n :

r = ] / m 2 + m 2 + m 2 , s : m 3 , a = a r c t a n ( m 2 / m l ) ( 3 . 3 a )a n d ~ i s i m p l i c i t l y d e f i n e d v i a

r 2 n a = a b s - a r ~ c o s ~ ~ ~ : ~ ( r , s , n a ) . ( 3 . 3 b )T h u s w e h a v e

w h e r e

G =

- - c o s a - - s i n a 0 0 0 0r r0 0 1 0 0 0 I

I1 1- - - : - - s i n a - - - c o s a 0 0 0 0 '1 'F 1 " I/- T & c o s a - -Q ~r s i n a - - ~ r + ~ 2 0 0 ~3 i

~ ~ ~ r 2P l ~ 8 -~ -, ~ 2 = ~ s a n d ~ a - - 8 n a - - a r ~ s i n

( 3 . 4 )

(3.5)


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Complex Equ i l i b ri a o f Buck led Rods 327a r e o b t a i n e d f r o m ( 3 .3 b ) . W e s h a l l n o t r e q u i r e t h e e x p l i c i t e x p r e s s i o n s f o r ~ ta n d 9 2 .W r i t i n g t h e m a t r i x J o f (2 .1 2 ) e x p l i c it l y i n m , n c o o r d i n a t e s , w e h a v e

J-- - -O

0 - - m 3 /n2m 3 0 - m r

- m 2 r n 1 00 - n 3 n 2n 3 0 - n l

- - n 2 n t 0

0 --n 3 n 2 I .n a 0 - - n l- -n2 n t 0 (3 .5)

T h e n e w s y m p l e c ti c f o r m ~ is t h e n g i v e n b y t h e m a t r i xJab = G J G r , (3.7)

a n d a n e l e m e n t a r y c a l c u l a t i o n u s i n g ( 3 .4 ) - (3 . 6 ) a n d ( 3. 1) y ie l d s0 O O l ]0 0 1 0 (3.8)

J ~ , b = [ , 0 - 1 0 0I[ - 1 0 0 0

T h u s t h e r e d u c e d H a m i l t o n i a n s y s t e m c a n b e w r i t t e n---- Ja , b V~ H a,b( x) (3.9)

w here x = ( r , s , a , 9 ) r an d H a , b( r , s , (r, 9 ) = H ( r n ( r , s , a ) , n ( r , s , a , 9 ) ) - T h ee x p l i c it f o r m o f Y ~,b m p l i e s t h a t ( 3. 9) i s c a n o n i c a l w i t h ( s, a ) a n d ( r, 9 ) a s c o n j u g a t ep a i r s o f v a r ia b l e s .

T o a p p r e c i a t e t h a t t h i s r e d u c t i o n p r o c e s s s i m p li fi e s t h e p r o b l e m , w e c o n s i d e rt h e f o l l o w i n g S p ec ia l c a se s . F o r a r o t a t i o n a l l y s y m m e t r i c r o d w e k n o w t h a tH ( m , n ) i s o f t h e f o r m

H ( m , n ) = ,r 17 6 2 + m ~ , m 3 , n 2 q - n 2 , n 3 , m ~ n a + m z n 2 , m ~ n 2 - m2nl) (3 .10)(ef t ANTMAN & KENNEY [1981]) . N o w Ha,b i s g i v e n b y

; , ~ ( r 2 - s 2 , s , a 2 ( 1 - ( b s - r r _c o s 9 ) 2 ~ b s - 7 r c o s 9H a , b ( r , S , 9 ) - r 4 ] , a r 2 ~ ,\ k (3.11)~a - 2 a r r s in 9 )- ~ 2 ( b r + 7 2 s cos 9) , - -7 -

I

a n d i s t h e r e f o r e i n d e p e n d e n t o f a . T h u s a is a c y c li c v a r i a b le a n d s a c o n s t a n t o ft h e m o t i o n .O n t h e o t he r h a n d t he c as e a = 0 ( n = 0 ) le a ds t o

Ho ,b (r , s , ~ , 9 ) = H(7 " co s ~ , ~ cos ( r, s , O, 0 ,0 ) , (3 .12)a n d h e r e 9 is c y cl ic a n d r a c o n s t a n t o f t h e m o t i o n .


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3 2 8 A . M I E LK E & P . H O L M E ST o p e r f o r m e x p l ic i t c a l c u l a t i o n s w e r e s t r ic t o u r a n a l y s i s t o v e r y s m a l l m

a n d n . T h e n o n l y t h e lo w e s t o r d e r t e r m s o f H ( m , n ) = 89o q m ~ + n 3 - [ - dg(] m 1 3 ,I n l ] m [ , I n ] 2 ) a r e r e l e v a n t . W e u s e t h e s c a l i n g m - --- - O r b , n = e O 2 h w i t hIh i = : 1 a n d d e f i n e1/ -I ( ff t , h , e , 6 ) = - ~ - H ( O r h , e O 2 h ) = 89oqfft~ + eh 3 + ~ ( ~ ) . ( 3 . 1 3 )

T h e c o r r e s p o n d i n g ( a , b , r , s , a , Q ) - sc a li n g i sa : e O2fi, b : E ~ 319, r = 0 k , s = ~ , ( a = b , e - : b ) , ( 3 . 1 4 )

w h e r e h = 1. H e n c e f o r t h w e n e e d t h e ( r, s , a , ~ ) - e o o r d i n a t e s i n t h e s c a le d v e r s i o no n l y . T h u s w e d r o p t h e h a t s o n ( r , s , a , p ) a n d b b u t r e t a i n t h e m o n ( r h , r~ ). W ed e f i n e

1

: 89 0r ~2 COS2 (7 2f_ 0~2;2 s in 2 a + 0r $2) ( 3 . 1 5 )E+ - ~ (b s - r ~ c o s e ) + r

O b s e r v e t h a t t h e l i m i t t5 ---- 0 i s i d e n t i c a l t o t h e h e a v y r i g i d b o d y i f e @ 0a n d n o t e t h a t e i s n o t n e c e s s a r i l y s m a l l ; t h is i s i m p o r t a n t i n o u r a n a l y s i s o f t h eD 2 v - s y m m e t r i c r o d .

I n b o t h e a s e s t h e c a n o n i c a l s t r u c t u r e o f t h e r e d u c e d s y s t e m e x p l ic i tl y re v e a l st h e s y m m e t r i e s i m p l i c i t i n t h e o r ig i n a l n o n - c a n o n i c a l s t r u c tu r e .

T h e c a s e o q < o~3 < o r r e q u i r e s a s l ig h t l y d i f fe r e n t c o o r d i n a t e s y s t e m . I nt h a t c a s e t h e u n p e r t u r b e d ( e = 0 ) b e h a v i o r i s e s s e n t i a l l y t h e id e n t ic a l to t h a t o f~ t < o~2 < ~ 3 w i t h m 2 , m 3 a n d n 2 , n a i n t e r c h a n g e d . I n s t e a d o f ( 3 .1 a , b ) ,w e d e f i ne a n a n a l o g o u s c o o r d i n a t e s y s t e m b y i n t e r c h a n g i n g m 2 a n d m 3 in d ef in i -t i o n ( 3 .1 a ) a n d n 2 a n d n a in (3 .1 b ) . E v e r y t h i n g g o e s t h r o u g h i n t h e s a m e m a n n e r a sp r e v i o u s l y , e x c e p t t h a t t h e H a m i l t o n i a n ( 3 .1 5 ) i s r e p l a c e d b y

/fib = 89 0hi ~2 CO S2 O" -~- 063r 2 s in 2 tr + 062$ 2 )+ ~ 22 ( ( b k + r s c o s ~ ) s i n a + r r c o s a s i n ~ ) + r ( 3 . 1 6 )

I n t h e r o t a t i o n a l l y s y m m e t r i c c a s e w e u s e t h e s a m e s c a l in g b u t w i t h e ---- I . H e r e ,s i n c e 0 q = or 2 , f o r e = 0 t h e i n v a r i a n t s p h e r e s [ r n 1 = t~ a r e f i l le d w i t h p e r i o d i co r b i t s l y i n g i n t h e p l a n e s m 3 = c o n s t . T o g e t a h o m o c l i n i c s o l u t i o n w e f ix e = 1a n d t r e a t t h e l i m i t ~5 = 0 , w h i c h i s e x a c t l y t h e c a s e t r e a t e d i n H O L M ES & M A R S D EN[ 1 9 8 3 ] .

4 . T h e M a i n R e s u l ts : E q u i li b ri u m S t a t e s o f B u c k l e d R o d sW e s t a t e o u r m a i n r e s u l t i n t e r m s o f t h e t w o - d i m e n s i o n a l s y s t e m s o n t h e r e d u c -

e d p h a s e s p a c e s P a ,o = ( ( m , n ) [ ] n I ---- a , m . n = a b ) :


11/30

C o m p l e x E q u i l ib r i a o f B u c k l ed R o d s 3 2 9Th e o r em4 . 1 . L e t H ( m , O ) = 8 9 q - d?(Im,[3). W e t h en h a v e t h e f o l l o w i n gcases :

I . 0 < ~1 < o~2 < o~a. Fo r a l l su f f ic ien t ly sm al l e , O > 0 an d a l l b E ( - 1 , 1 )the r e e x i s t s a pa i r o f pe r iod ic s o lu t ions (m , n ) = (0 , -q -0 , 0 ; 0 ) q - d ~ (0 2) o nP~o,,~n~b A ( H = 89 ~2 02} whic h a r e c onn e c te d by t r ans v er s e he te r oc l in ic c y c le s.

II . 0 < oq < o~ < o~2. For a l l s u f f ic i e n t l y s m a l l e , ~ ~ 0 and a l l b s a t i s f y ing9-g20rb 2 < (4.1)

~ 2 0 ~ 2 - i t - 4 ( 0 ~ 2 - C ~ 3 ) (~ 3 - 0 ~ 1 ) c o s h 2 ( Y t~ 16 2 )2 I / (g2 - ~3) (~3 - ~t h e re e x i s t s a p a i r o f p e r i o d i c s o l u ti o n s (m , n ) : (0 , 0 , 4 -0 ; 0 ) q - dT(0 2) onP~,,~n~b A ( H : 89 ca 02} whic h a r e c onne c te d by t r ans v e r s e he te r oc l in ic c y c le s .

T h e s e c o n d r e su l t c o n c e r n s ro d s w h i c h a r e s y m m e t r i c w i t h r e s p e c t t o r o t a t i o n st h r o u g h t h e a n g l e 2 z l / N a b o u t t h e i r a x e s i n t h e r e f e r e n c e c o n f i g u r a t i o n . I n t h e( r , s , (7, ~ ) - c o o r d i n a t e s t h i s c o r r e s p o n d s e x a c t l y t o t h e t r a n s f o r m a t i o n a - - ~ (7 +2 ~ ] N . M o r e o v e r w e a s s u m e t h a t t h e c r o s s - s e c t i o n h a s a l s o a r ef l e c ti o n a l s y m m e t r y ;a n d w i t h o u t l o ss o f g e n e r a l i t y l e t t h e d 2 - a x i s b e t h e s y m m e t r y a xi s. T h e n t h eH a m i l t o n i a n H - - - - H ( m , n ) i s i n v a r i a n t u n d e r t h e r e f l e c t i o n s S l : ( m , n ) - +( m l , - m 2 , - m 3 , - n l , n2 , na ) a n d S 3 ( m , n ) -- ~ ( m l , m 2 , - m a , - n l , - n 2 , n 3)w h i c h c o r r e s p o n d t o r e f l e ct io n o f m a t e r ia l p o i n t s i n t h e r o d w i t h r e s p e c t t o t h ed 2 , d a - p l a n e a n d d r , d 2 - p l a n e r e s p e c ti v e l y . O b s e r v e t h a t o n l y t h e c o m p o s i t i o nS ~ - S1Sa m a p s P a,b o n t o i t s e l f i f b ~ 0 . S i n c e S : ( r , s , (7, ~ ) - ~ ( r , s , - ( 7 , - ~ )t h e H a m i l t o n i a n / t b s a t i s f i e s

/-Ib(r, s , (7, ~, e, 0 ) = ~Ib(r, s, ~ + 2z~/N, ~, e, O) = I~b(r, S, --(7, - -~ , e, O). (4 .2 )T o f i nd t h e l o w e s t o r d e r a t w h i c h t h e f ir st n o n t r i v i a l D N - s y m m e t ri c t e r m c a no c c u r w e h a v e t o a p p e a l t o t h e m e t h o d s i n i n v a r i a n t t h e o r y ( c f. BU Z AN O , G E Y-M O NA T, & P O S T O N [ 1 98 5 ]) . I n o u r c a s e t h e c o r r e s p o n d i n g p o l y n o m i a l s , b e i n gD s - i n v a r i a n t b u t n o t r o t a ti o n a l l y s y m m e t r i c , a r e o f t h e f o r m R e ( m l q - i r a 2 ) k(n I -~- in2) N - k w h e r e k ---- 0 , 1 . . . . N . H o w e v e r , s in c e o u r s c a l in g is o f t h e f o r mm : O r b , n = O 2 h, ( r e c a ll e : 1 h e r e ) t h e l o w e s t o r d e r t e r m i s R e (m l + ira2)Nwi th o rd e r d T(0 N) .

F o r N => 3 a n d u n d e r t h e a s s u m p t i o n t h a t H = H ( m , n ) i s N + 1 t im e sc o n t i n u o u s l y d i f f e re n t i a b le , w e c o n c l u d e t h a t t h e s c a l e d H a m i l t o n i a n h a s t h e f o r m

I ~ b ( r , S , ( 7 , Q , e , O ) : I - I ~ ( r , s , Q , e , O ) - ~ - c 0 N - - 2 ( r 2 - - 3 2 ) N I 2 C O S N o " -+ - d g ( 0 N - I ) .(4 .3 )

O b s e r v e t h a t t h e q u a d r a t i c t e r m s a r e ( 7 - in d e pe n d en t , i .e . H ( m , n ) = 89 ~(m 2 ~- m 2)-t- oc3m2) -t- tP(] m 13) a n d th u s c~1 = 0~ = o~ ~ o~3. W i t h o u t l o ss o f g e n e r a l i t yw e r e s t ri c t o u r a t t e n t i o n t o t h e c as e e = 1 .W e c a n n o w s t a t e t h e s e c o n d m a i n r e s u l t :Theorem 4 . 2 . L e t t h e H a m i l t o n i a n s a t i s f y (4 .2 ) w i t h N ~ 3 a n d a s s u m e t h a t cin (4 .3 ) i s nonz e r o . T he n , f o r a l l s u f f i c ie n t l y s m a l l O > 0 and a l l b w i th 0 < b 2 < 4]o~


12/30

3 3 0 A. MIELKE & P. HOLMESt h e r e e x i s t s a p e r i o d i c s o l u t i o n m = ~ b e 3 + r n = 62 e a + d)(O ) o nP ~ 2,~ b A H = ~ 2 1 q - - ~ b 2 w h i c h p o s s e s s e s t r a n s v e r s e h o r n o c l in i c o r b i t s .

T o a p p r e c i a t e t h e p h y s i c a l i m p l i c a t i o n s o f t h e s e r es u lt s, w e m u s t a n t i c i p a t es o m e o f t h e m a t e r i a l o u t l i n e d i n S e c t io n 5. T h e t h e o r e m s a r e p r o v e d b y p e r t u r b a -t i o n a r g u m e n t s i n v o l v i n g s o lu t i o n s l y in g c lo s e to h o m o c l i n i c o r h e t e r o c l in i c o r b i tst o h y p e r b o l i c s a d d l e p o i n t s o f th e u n p e r t u r b e d s y st e m s . T h e e x i s te n c e o f t ra n s v e r s eh o m o c l i n i c o r h e t e r o c l i n i c p o i n t s t h e n i m p l ie s , v i a t h e S m a l e - B i r k h o f f h o m o c l i n i ctheorem (GtJCKENHEIMER& HOLMES [1983, w5 . 3] ) , t h a t t h e r e e x is t s o l u t i o n s w h i c hr e m a i n i n a n e i g h b o r h o o d o f th e u n p e r t u r b e d h o m o c l i n i c o r h e t e ro c l in i c o r b i tsa n d w h i c h a r e c h a o t i c i n t h e f o l l o w i n g s en s e. T h e r e a r e t w o ( o r m o r e ) d i s j o i n tc l o s e d s et s i n t h e p h a s e s p a c e w h i c h t h e s o l u t io n s p a s s t h r o u g h i n a n y p r e s c r i b e ds e q u e n c e . T h e s e t s c a n b e c h o s e n t o l ie in a n y n e i g h b o r h o o d o f t h e s a d d l e p o i n t ( s ) ;i n t h e e v e n t o f a s e t o f m u l t i p le h o m o c l i n i c o r b i t s o r h e t e r o c l i n i c c y c le s , t h isi m p l i e s t h a t t h e p e r t u r b e d s o l u t i o n p a s s es n e a r d i f f e r e n t m e m b e r s o f t h e s e t ina n y o r d e r . T h e c o n s e q u e n c e is t h a t n o n p e r i o d i c o r b i ts n e a r t h e t ra n s v e rs e h o m o -c l in i c o r h e t e r o c l i n i c o r b i t s a r e , t o f i rs t o r d e r , q u a s i - r a n d o m s u p e r p o s i t io n s o fs in g le , u n p e r t u r b e d h o m o c l i n i c o r h e t e r o c l in i c o r b i ts . T h u s , t o u n d e r s t a n d t y p i c a lg l o b a l s t r u c t u r e s o f p e r t u r b e d o r b i ts , a n d t h e s p a t i a l e q u i l i b r i u m s t at e s t o w h i c ht h e y c o r r e s p o n d , w e m u s t f i r st c o n s i d e r t h e s p a t ia l s t a te s c o r r e s p o n d i n g t o u n -p e r t u r b e d o r b i t s .

A s w e o b s e r v e d a t t h e e n d o f S e c t i o n 2 , t o o b t a i n e q u i l i b r i u m s h a p e s f r o m t h es t re s s e s ( r e ( t ) , n ( t ) ) , w e m u s t i n t e g r a t e e q u a t i o n s ( 2 . 1 7 a , b) a n d r e c a ll t h e o r i g i n a ls p a t i al d e s c r i p t i o n o f t h e d e f o r m e d r o d o f e q u a t i o n ( 2. 1) . W e w i ll c o n c e n t r a t e o nt h e i m p l i c a t io n s f o r t h e v e c t o r r ( t ) d e s c ri b i n g t h e p o s i t i o n o f t h e a x is o f t h e d e f o r m e dr o d . T o a r r a n g e t h e d i s cu s s io n c l e a r l y w e w i l l o n l y d e a l w i t h t h e s c a li n g l im i t

= 0 . T h u s w e s c a le r ( t ) a n d R ( t ) i n t h e f o l l o w i n g w a y1r ( t ) - ~ - - ~ ~'(t~t) , R ( t ) ~ R ( a t ) . (4 .4 )

F o r ~ = 0 w e a r e l ef t w i th

i f r h ( t ) i s a l r e a d y k n o w n .

0 -o~3r~3 ~rh2or 0 - or 1/~/1

-o~2r~2 or 0( 4 . 5 a , b )

In th e c a s e 0 < ~ < oc2 < or 3 t h e f o u r h e t e r o c l i n i c s o l u t io n s o n t h e m a n i f o l dI , h l = 1 , I h i = 0 a r e g i v e n b y

rna( t ) ~ = | - K , K 2 ta n h (~/a -~a3 t ) (4 .6 )r h 3 ( t) Jl I K 2 1 / - a 3 I a 2 s e c h ( I / a - -~ - ~ )w h e re K 1, K 2 E { - 1 , 1 } , a 1 = o r a 2 = o ~ i - 0 ~ a < 0 , a n d a3----o~2- 0 q > 0 . I t w i ll b e c o n v e n i e n t t o e x p r e ss t h e r o t a t i o n m a t r i x / ~ i n t e r m s o f t h e


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Complex Equ i l ib r i a o f B uck led Ro ds 331c lass i ca l E u le r ang les ~ , 0 , q0:w h e r e lq = D~ Do D,p , (4 .7a )

COS~p - s in ~p 0 ] 1 0 0 ]1~ = sin ~p c os ~o 0 , / ) 0 = 0 c o s 0 - s i n 0 . ( 4 .7 b )0 0 1 ] 0 sin 0 co s 0S i n c e F = / } r l - - - - 0 f o r t h e h e te r o c l in i c o r b i ts , w e h a v e M - - - - / } r h = c o n s t .W e a r e f re e t o o r i e n t o u r s p a t i a l c o o r d i n a t e s a n d , t h e r e f o r e , w e p i c k M = e 3s i n c e t h i s s im p l i f ie s th e c a l c u l a t i o n s . U s e o f t h e d e f i n i t i o n ( 4. 7) y i e l d s t h e n

s i n 0 s i n g ] .t h = / i r e 3 = s i n 0 c os ~o [ (4 .8 )

c o s 0C o m p a r i n g t h i s e x p r e s s i o n w i t h ( 4 . 6 ) , w e f i n d

cos 0 = rh3, 9 = ar ct an ( rh l / rh2) . (4 .9)W e w i ll a ls o r e q u i r e ~p. F r o m ( 4 . 5 b ) a n d (4 .7 ) w e d e r i v e , a f t e r s o m e c a l c u l a t i o n ,t h e r e l a t i o n

,p' sin 0 = ~ i r h ~ in 9 + 062/~ 2 COS 99~ (4.10)or , wi th (4 .8) and (4 .9) ,

0~l/~/l @ 062/~/2~ / ) ' = ~ 1 s in 29 - / a 2 cos29~ - - rh 2 + if72 (4 .11)

A s s u m i n g w i t h o u t lo s s o f g e n e r a l i t y t h a t ~ p(0 ) = 0 , b y d i r e c t i n t e g r a t i o n w e o b -t a i n~0(t) = 0r - ar ct an ( ] / a 3 / a l t a n h ( t / a - ~ 3 t ) ) . ( 4 .1 2 )

W e a r e n o w i n a p o s i ti o n t o c o m p u t e t h e e q u i l i b r iu m s h a p e i n t e r m s o f t h ed i s p l a c e m e n t v e c t o r ~ ( t ) b y i n t e g r a t i n g ( 4 . 5 a ) . U s i n g ( 4 . 7 ) w e o b t a i n

s in 0 s in ~ ot= ~ ( 0 ) + f - s i n 0 c os~p0 c o s 0

R ea l iz in g th at s = rh3 = co s 0 , s in 0 = 1/1 - s 2 ,w e u s e t h e r e l a t io n s ( A . 5 a , b ) t o o b t a i n

1 V - - a t a 2r l ( t ) -~ r l (0 ) - - - s in 0 cos ~p +0r 2 0r 2

l / - - a i a 2r2( t ) = r2(0) + 1 s in 0 s in ~ +0r ~ 2

2ra( t ) = r3(O) + K2 i / _ a i a 2

d t . (4 .13)

a n d ~ o( t) = - ~ ( t ) ( c f . (6.2)) ,t

f COS 2 0 s i n or d t ,0

tf COS2 0 COS 0r d t ,

0

- - arctan ( tanh ( ] /a- -~ 'a t /2 ) ) .

(4 .14)


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33 2 A. MIELKE & P. HOLMEST o u n d e r s t a n d t h e g e o m e t r i c s h a p e o f t h e s e s o l u t i o n s w e f ir s t l o o k a t t h e

l im i t s t - + ~ o ~ . S i n ce c o s 0 ( t ) - - - > 0 a n d ~ o ( t ) - ~ 2 t - - - ~ z A 1 w i t h A I ~a r c t a n (r w e h a v e f o r t - + -q- o o

[ A2 cos (~ ,2 t - 4 -A 0I s in (~x2t -q-A~) -5 d~(e-f~-~ l t l ) , (4 .15 )1~(t) = ~(0) + ~ A 3 + a-~2[ ~ I ~ A , o

w h e r e A z i s s o m e c o n s t a n t a n d A 3 a n d A 4 a re g i v e n b y_ ~ [ ~ t ~ 2 ~ 1A3 2 " - a ~ a 2 c ~' ( - ' VV a ~ a a ] c ~ 2 A 4 = ~ / 2 ~ / - a ~ a 2 " (4 .16)

T h e o s c i l l a t o r y b e h a v i o r f o r l a r g e t c o r r e s p o n d s t o t h e f a c t t h a t t h e s a d d l e p o i n t st h ---- (0 ~, I , 0 ) , f i ~ 0 , r e p r e s e n t r o d s i n p u r e b e n d i n g , c o i l e d o n t o t h e m s e l v e si n o n e p l a n e , h e r e p a r a l l e l t o t h e I , 2 - p l a n e . T h e f u l l r o d c o n f i g u r a t i o n , o b t a i n e db y n u m e r i c a l i n t e g r a t i o n o f ( 4. 14 ) f r o m t ~ - 1 0 t o 1 0, i s s k e t c h e d in F i g u r e 3 .

9 A z r ~

F i g . 3. S k e t c h o f t h e r o d c o n f i g u r a ti o n c o r r e s p o n d i n g t o t h e u n p e r t u r b e dhete rocl in ic orb i ts for 0 < ~1 = 2 -- 1/1/3-< c~2 = 2 < a3 = 2 + ] /3- .

A s F i g u r e 2 i n d i ca t e s , o n t h e s p h e r e I th i = 1 t h e r e a r e f o u r h e t e r o c l i n i cs o l u t i o n s . Y e t w e o b t a i n o n l y t w o d if f e re n t g e o m e t r i c s h a p e s ( K : = ~ 1 ) . T h ed i f f e r e n c e b e t w e e n t h e c a s e s K 1 ---- + I a n d K t = - 1 i n ( 4 .6 ) i s o b t a i n e d b yc h a n g i n g ~ t o q~ + ~ , i.e. t h e c e n t e r l in e s t a y s t h e s a m e b u t t h e r o d i s f ir s t r o t a t e db y re a r o u n d i ts c e n t e rl i n e a n d t h e n b e n t i n t o t h e s a m e c o n f i g u r a ti o n .

T o o b t a i n a n i d e a o f w h a t t h e c h a o t i c e q u il i b ri u m s ta t es o f s u c h a r o d m a yl o o k l i k e , w e r e m i n d t h e r e a d e r t h a t s u c h s o l u t i o n s a l w a y s r e m a i n n e a r o n e o ft h e p e r t u r b e d h e t e r o c l i n i c s o l u t i o n s . T h e s e a r e t h e m s e l v e s c l o s e (d ~(e )) t o t h eu n p e r t u r b e d s o l u t io n s . H e n c e w e m a y t h i n k o f t h e c h a o t i c s t a te s a s f a i rl y a r b i t r a r yc o m b i n a t i o n s o f " e l e m e n t s " , e a c h o f w h i c h is , u p t o r a n u n p e r t u r b e d s o lu t i o nt a k e n o v e r a l a r g e b u t f in i te a r c l e n g t h . O f c o u r s e w e h a v e t o s a t i s fy t w o c o n d i t i o n s .F i r s t t h e r e is a " d y n a m i c a l " c o n d i t i o n t h a t n e a r a s a d d le p o i n t t h e r e a r e o n l y th r e ec h o i c e s f o r t h e c o n t i n u i n g s o l u t i o n : e i t h e r it r e m a i n s t h e r e o r l ea v e s i n th e n e i g h -b o r h o o d o f o n e o f t h e t w o b r a n c h e s o f t h e u n s ta b l e m a n i f o l d . S e c o n d , t h e re i s


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C o m p l e x E q u i l ib r ia o f B u c k le d R o d s 3 3 3t h e o v e r a l l c o m p a t i b i l i ty o f t h e e q u i l i b r iu m t o c o n s id e r . W e m u s t a r r a n g e e a c h" ' e l e m e n t a r y " c o n f i g u r a t i o n i n N 3 i n s u c h a w a y t h a t t h e r e s u l t a n ts F a n d Ma r e t h e s a m e a s f o r t h e o t h e r e l em e n t s .

T h e t w o o t h e r c as es , 0 < 0 q < 0 ~ 3 < 0 ~ 2 a n d 0 < 0 ~ = 0 ~ 2 < o c a , l ea d i na s im i l a r fa s h i o n t o t h e c o r r e s p o n d i n g s h a p e s o f h e t e r o c li n i c o r h o m o c l i n i c s o l u -t i o n s i n t h e u n p e r t u r b e d c a s e . F o r 0 < o q < 0r 3 < 0r 2 t h e f o u r h e t e r o c l i n i cs o l u t i o n s a r e

K x 1 / - a l /a a s e c h ( [ /a - ~ - 2 t ) ]/r h ( t) = K z I / - a 2 / a 3 s e c h ( I / a - - ~ t ) | . ( 4 . 1 7 )/K 1 K 2 t a n h ( r t ) J

F r o m ( 4 .8 ) w e o b t a i n c o s 0 ---- K 1 K 2 t a n h ( l /a l a 2 t ) a n d ~0 ---- K I K 2 a r c t a n l /aa/a2.H e n c e , (4 . 1 1 ) y i e l d s ~ o ( t ) - -- - s a t a n d ( 4 . 1 3 ) g i v e s

s e ch ( ] / a - ~ t ) s i n ~ a tt~'(t) = ~(0) + f - s e c h ( l / a - ~ t ) c o s o r d t .0 K 1 K z t a n h ( i / a - ' ~ t )

I n t h e l i m i t t---> - 4 - o o ~ ( t) b e h a v e s a s fo l l o w sB I 0

~( t ) --= ~(0 ) + - -~-B + K IK z 0 + d~(e v~ -~ l t l )~ B 3 t

( 4 . 1 8 )

/ ~ ' ~ 3w h e r e B 2 - - s e c h |2 [ /a la2 ~2 ] /a - -~ ] "I n t h e c a s e 0 < o q = o~2 = o~ < o~3 th e u n p e r t u r b e d s o l u t i o n i s a h o m o c l i n i c

s o l u t i o n d e f i n e d b y

( 4 . 19 ))( t ) = - ~ 3 b t, c o s e = ~ r - 1 ,4w h e r e 7 2 = - - - b 2 . N o w F = R f i is d i f fe r e n t f r o m z e r o , a n d w e m a y a s s u m e

F = e 3. I n a m a n n e r s i m i l a r to ( 4 .8 ) w e o b t a i n b y u s i n g ( 3 .1 )/,7 7- - co s ~ ( 1 + co s ~ ) - - - s i n ~ s i n 0 s i n 0 s i n ~0r 2 rb r 7h = ~ - s i n o ( 1 + c o s ~ ) + - - r c o s ~ s i n 0 = s i n 0 c o s ~

b 2 ~2r 2 r 2 c o s Q c o s 0


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3 3 4 A. MIELKE & P. HOLMESH e n c e , ( 4 . 1 9 ) g i v e s u s

c o s O = l - T s e c h 2 t .U s i n g ( 4 .1 3 ) a g a i n , w e f i n d t h a t t h e f o r e g o i n g im p l i e s f o r l ' ( t) t h e l i m i t b e h a v i o r

- 4 - c o I~ ' (t ) = Cz + 0 " + ~ t e - - s (4 .20)

/ a~, \

Itf o r t - + ~ ~ . I n th e s e t w o c as es t h e a s y m p t o t i c s ta te s ( c o r r e sp o n d i n g t o th es a d d l e p o i n t s i n ( m , n ) - s p a c e) a r e s t r a i g h t r o d s ; i n ( 4. 18 ) i n p u r e t o r s i o n a n d i n( 4 . 2 0 ) i n m i x e d t o r s i o n a n d t e n s i o n .

5 . S e c o n d R e d u c t io n : M e l n i k o v ' s M e t h o d

I n t h i s s e c t i o n w e o u t l i n e t h e m e t h o d o f M E L NIK O V [ 1 96 3 ], a s g e n e r a l i z e d a n da d a p t e d t o t h e a n a l y si s o f H a m i l t o n i a n s y s te m s h a v i n g t w o d e g r e e s o f f r e e d o mb y H O LM E S & M A R SD E N [ 1 98 2 , 1 9 8 3 ]. W e f i r st o u t l i n e t h e r e d u c t i o n p r o c e d u r e( o f . BIRKHOFF [1927 , Ch . 8 ] , WH ITTAKER [1937 , Ch . 12], ARNOLD [1978 , w 4 5 B]) .

W e s t a r t w i t h a H a m i l t o n i a n o f t h e f o r mH e = H o ( q , p , I ) + e H l ( q , p , O , I , e ) , (5 .1 )

w h e r e ( q , p ) a r e c o n j u g a t e v a r i a b l e s a n d ( L 0 ) a r e c o n j u g a t e v a r i a b l e s i n a c t i o n - a n g l ef o r m , s o t h a t H t i s 2 z ~ -p e ri o di c i n 0. F o r e = 0 0 i s a c y c l i c v a r i a b l e a n d H a m i l -t o n ' s e q u a t i o n s a r e c o m p l e t e l y i n t e g ra b l e . W e s u p p o s e H ~ ( q , p , 0 , / , 0 ) i s b o u n d e do n b o u n d e d s et s a n d t h a t H o a n d H I a r e s u f f ic i en t ly d i f f er e n t ia b l e f o r t h e p o w e rs e r ie s m a n i p u l a t i o n s w h i c h f o l l o w ( C a w i l l s uf fi c e) .

O u r s pe ci fi c a s s u m p t i o n s o n t h e u n p e r t u r b e d H a m i l t o n i a n H o ( q , p , I ) a r e t h a t( 1 ) T h e s y s t e m

i l = c~Ho/6p, b = --~3Ho/c~q (5 .2 )p o s s es s e s a h o m o c l i n i c o r b i t x h = ( q ( t - t o ; h ) , ~ ( t - t o ; h ) ) t o a h y p e r b o l i cf i x e d p o i n t X o = ( q o, P o ) f o r e a c h t o t a l e n e r g y H o = h i n s o m e i n t e r v a l J ~ R .N o t e t h a t ~ h d e p e n d s o n h v ia th e a c t i o n I = Ih c o r r e s p o n d i n g t o t h e h o m o c l i n i co r b i t a n d t o t a l e n e r g y : H o ( ~ h , Ih ) = h .

( 2 ) F o r h E J a n d (q , p ) ~ -Xh(t - - to ) , t h e f r e q u e n c y~ H oS ~o - ~ I ~ o ( ~ h ( t - t o ) , I D ( 5 . 3 )

o f t h e u n p e r t u r b e d s y st e m Ho(-Xh, Ih) ~- h satisfie s [Qo i ~ 6 > 0.U n d e r t h e s e h y p o t h e s e s , w e m a y i n v e r t t h e e q u a t i o nH , ( q , p , O , I ) = h E J (5 .4 )


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C o m p l e x E q u i l i b r i a o f B u c k l e d R o d s 3 3 5i n a n e i g h b o r h o o d o f t h e u n p e r t u r b e d h o m o c l i n i c o r b i t a n d s o l v e f o r th e a c t i o n Iha s a f u n c t i o n o f ( q , p , 0 ; h ) :

Ih = J , ( q , p , 0; h) = or + eJ r + d~ (e2 ). ( 5 . 5 )S H xM o r e o v e r , h y p o t h e s i s (2 ) i m p l ie s t h a t t h e e q u a t i o n d O / d t = D e = D o + e S lc a n b e i n v e r t e d f o r s m a l l e , a n d h e n c e t h a t " r e a l " t im e t c a n b e r e p l a c e d b y t h e

a n g l e 0 t o g i v edq :dq dO SHe de t sH e~ --~ -- ~ - / - ~ - = . Q Z 1 ~ p , ~ - ~ = - ~Q Z ~ q 9 (5 . 6 )

I m p l i c it d i ff e r e n t i a ti o n o f H e = h y i e l d sS H , D , S : e = 0 S H e + D e c % r~---~ + S q = - ~ p ~ p p , (5 . 7 )

s o t h a t, f r o m ( 5 .6 ) - (5 . 7 ), w e o b t a i n t h e r e d u c e d e q u a t i o nq , _ s J e s oa eS p (q ' p ' 0 ; h ) , p ' = ~ (q , p , 0 ; h ) (5 . 8 )

do n e a c h e n e r g y s u r f a c e H e = h E J . H e r e ( ) ' d e n o t e s ~--~ ( ) , d i f f e r e n t i a ti o n w i t hr e s p e c t t o t h e a n g l e v a r i a b l e , w h i c h p l a y s t h e r 6 1 e o f t h e n e w t im e .T h e s e ri es e x p a n s i o n ( 5 .5 ) f o r J e c a n b e c o m p u t e d d i r e c tl y b y e x p a n d i n gH o + ~ H , = h w i t h I r ep la c ed b y J o + e J ~ + . . . O n e o b ta in s

J o = ~r P ; h ) = H o ( q , p ) - i ( h ) , ( 5 . 9 a )- H t ( q , p , O , H 6 q ( q , p ) ( h ) )or 1 = or q , p , 0 ; h ) = D0 (q , p , H o t ( q , p ) ( h )) ' ( 5 . 9 b )

w h e r e H o ( q , p ) - 1 ( h ) d e n o t e s i n v e r s i o n o f r i o w i t h r e s p e c t to t h e v a r i a b l e L T h u s( 5 .8 ) t a k e s t h e f o r m o f a p e r io d i c p e r t u r b a t i o n o f a n i n t e g r a b le H a m i l t o n i a ns y s t e m : s p e c i f i c a l l y , f r o m ( 5 . 7 ) a n d t h e e x p a n s i o n ( 5 . 5 ) , ( 5 . 9 ) , w e h a v e

& C o SJ 1q ' - - ~ - -s ? s ? + r ( 5 . 1 0 )

& C o SJ 1p ' = + ~ - -S q S q( S : o S : o l S U o S H oa n d t h e u n p e r t u r b e d v e c t o r f ie l d , - S p ' ~qq ] is s i m p l y D o t \ ' ~PP , S q ] ' a

s c a l e d v e r s i o n o f t h e u n p e r t u r b e d f ie l d o f t h e o r i g i n a l p r o b l e m , r e s t r i c te d t o ( q , p )s p a c e . T h u s h y p o t h e s i s ( 1 ) i m p l i e s t h a t , f o r e = 0 , ( 5 .1 0 ) h a s a h o m o c l i n i co r b i t t o a h y p e r b o l i c f i x e d p o i n t .T h e s t a n d a r d M E L N IK O V m e t h o d a s d e v e l o p e d i n G U C K EN H EIM E R H O LM E S[ 1 9 53 , w 4 .5 ] c a n b e a p p l i e d d i r e c t l y t o ( 5 .1 0 ) . W e h a v e


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336 A. MIELKE & P. HOLMESP r o p o s i t i o n 5 . 1 . L e t X h : ~ h ( 0 - - O o ) , P h ( 0 - - 0 o ) d e n o te t h e h o m o c l i n ic o r b i tt o t h e f i x e d p o i n t P o o f t h e u n p e r tu r b e d H a m i lt o n i a n s y s t e m J o ( q , P ; h ) in t h e e n e r g ysur f ace H~ : h an d de fi ne t he M e l n i ko v f unc t i on

ooih(Oo) = f { J o , J , } ~ h ( 0 ) , ~ h ( 0 ) , 0 + 0 o ) dO. (5 .11)--coThe n f o r e ~= 0 su f f ic i en t ly smal l , i f Mh(Oo) has s i mpl e z e ros , t he s t ab l e and uns t ab l em a n i f o l d s o f th e p e r t u r b e d f i x e d p o i n t p , = P o + ~ (e ) o f t h e P o in c ar 6 m a p c o r-respond i ng t o (5.8) i n te r s e c t t ra n s v e rs e ly f o r t h e p e r t u r b e d s y s t e m J , . I f M h (O o )i s bou nde d aw a y f ro m zero , t hen t he man i f o l ds do no t i n te r sec t.Proof. See GREENSPAN & HOLMES [1982] or GUCKENHEIMER ~r HOLMES [1983] .Th e m ain ideas an d or ig ina l pr oo f a re d ue to MELNmOV [1963] (cf . ARNOLD[1964]).

I t is unne c e s s a ry t o c om pu t e J o a nd or e xp li c it ly , f o r w e ha veL e m m a 5 .2 . (HOLMES~r MARSDEN [198 3])

} { H 1} (~h(t), ~h(t) , ih, ~(t) + Oo) dt ' (5.12)M h ( O 0 ) = - o o H ~ ~ o (q,p )wh ere { ', "}(q ,p) den otes tha t on ly the var iables (q , p) are us ed in the bra cke t evalua-t io n . I h i s t h e ( c o n s t a n t ) a c t io n g i v e n b y H o ( q , p , I h ) = h a n d - O ( t ) =tf / 2 ( ~ ( s ) , ~ h ( s ) , I h ) ds.0P r o o f . C ons i de r t he e qua t i on

H o ( q , p , J o + eJ,) + e H l ( q , p , O , Jo + e J ~ ) = h q - d?(e2);th i s impl ies tha t

~ H o ~ J o e H o 0 J o (5 .1 3)~qq /20 Oq ~ - p - /20 ~pa s w e l l a s J l = - H 1 / / 2 o , as in (5 .9b) . Therefore

{ J o , ' f l } --cO ~ &-r cl J o el ,~q ~p 8p Oq

1 ~?H o c9 1 ~?H o r(H~l/2o) + ( - H, I /2o)/2o ~q ~P /2o ~P ~q1 ( H 1 }

= (5.14)S ince dO = / 2 d t, sub st i tut ion o f (5.14) in (5.11) yields (5.12) . [ ]


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Co m p le x Eq u i l ib r i a o f Bu c k le d Ro d s 3 3 7P r o p o s i t i o n 5.1 a n d L e m m a 5 .2 t o g e t h e r y i e ld :T h e o r e m 5 . 3 . I f H , : - H o - ~ e l l 1 s a t is f ie s h y p o t h e s e s (1 ) a n d ( 2 ) a n d t h e M e l n i k o vf u n c t i o n M h ( O o ) o f (5 .1 2 ) has s imp le zeros , t hen, f o r e ~ 0 su f f ic i en t l y sma l l ,t here e x i s t t ransverse homocl in i c o rb i t s t o a hyperbo l i c per iod ic o rb i t on the energysur face H~ - : h .T h e S m a l e - B i r k h o f f h o m o c l i n i c t h e o r e m (SMALE 1963], [1967], GUCKENHEIMER& HOLMES [1 9 8 3 ] ) th e n imp l ie sC o r o l l a r y 5 . 4 . The P o incar$ ma p a ssoc ia t ed w i th H , on the l eve l se t H~ - l (h ) hasa hyperbo l ic , non -wander ing Can tor se t Oh on wh ich the map i s con juga te t o a sub -s h i f t o f f i n i t e t y p e .A s M O S E R [ 1 97 3 ] s h o w s , t h i s i n t u r n i m p l i e sC o r o l l a r y 5 .5 . 1-18possesses no ana ly t i c i n tegra l s o f mo t ion independen t o f t he to ta lenergy H8 i t sel f .

I n t h e f i r s t t w o s i t u a t i o n s t r e a t e d i n t h i s p a p e r , r a t h e r t h a n a h o m o c l i n i c o r b i tt o a f i xe d p o i n t w e h a v e a c y c l e o f f o u r h e t e r o c l i n i c o r b i t s c o n n e c t i n g a p a i r o fs a d d l e p o i n t s (cf . F i g u r e 2 ). T h e t r a n sv e r s e h o m o c l i n i c o r b i ts o f T h e o r e m 5.3b e c o m e t r a n s v e r s e h e t e r o c l i n i c c y c le s , b u t o t h e r w i s e i t a n d t h e c o n c l u s i o n s o f t h ec o r o l l a r i e s s t a n d u n c h a n g e d . S e e H O L M E S & M A R SD E N [ 1 98 3 , F i g u r e 5 ] f o r a ni m p r e s s i o n o f t h e s t r u c t u r e o f s u c h c y c le s .

6 . P r o o f o f T h e o r e m 4 . 1T o p r o v e t h e t h e o r e m w e c o m p u t e M e l n i k o v f u n c t i o n s f o r s u i t a b l y s c a l e d

v e r s i o n s o f t h e H a m i l t o n i a n . S p e c i f i ca l l y , f o r c a s e I , oct < oc2 < o r w e t a k e(3 .1 5 ) a n d fo r c a s e I I , oc < oc3 < o c2 , (3 .1 6 ) . In p a r t i c u l a r we re s t r i c t th e c o m p u -t a t i o n s t o t h e l i m i t c a s e t~ = 0 ; t h i s s u f fi c e s s i n c e t h e d e p e n d e n c e o n t~ i s c o n -t i n u o u s. T h e c o n c l u s i o n s o f t h e t h e o r e m t h e n f o l l o w u p o n a p p l i c a ti o n o f T h e o -r e m 5 .3 . C e r t a i n c o m p u t a t i o n a l d e t a i ls a r e re l e g a t e d t o t h e A p p e n d i x .

W e r e m a r k t h a t t h e u n p e r t u r b e d H a m i l to n i a n s Ho(r , s , a ) d i f f e r o n l y i n t r a n s -p o s i t i o n o f oc2 a n d o c3 . I t t h e r e f o r e s u f fi ce s t o c o n s i d e r t h e u n p e r t u r b e d h e t e r o -c l in i c s o l u t i o n s o n l y t h e f ir s t c a se . T h e u n p e r t u r b e d H a m i l t o n ' s e q u a t i o n s m a y b ew r i t t e n

;2= - T / ~ ' ( ~ ) '

ir = s(fl(tr) - oca), (6 .1)k - - - - 0 ,

c l e fb = - r / ~ ( ~ ) = O ( r , ~ )


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3 3 8 A. MIELKE & P. HOLMESw h e r e 3 ( 0 ) = o q c o s s a + 0~z s i n s o a n d w e r e c a l l r 2 = r s - s 2 . W i t h o u t l o s so f g e n e r a l i t y w e f i x t h e u n p e r t u r b e d " m o m e n t u m " r = 1. i n w h i c h c a se o n e o ft h e f o u r h e t e r o c l i n i c o r b i t s c o n n e c t i n g t h e f i x e d p o i n t s ( s, o ) = ( 0, - 4- ~ /2 ) t a k e st h e f o r m

s = t/--'--'--'~3/azs e c h ( - I / a t a 3 t ) ,o ---- a r c t a n ( I / ~ s in h ( - I/a--~3 t ) ) , ( 6 .2 )

= - [ a 2 t + a r c t a n ( I /a 3 - - ~ t a n h ( - t /a~a3 t ) ) ] ,w h e r e a t = 0% - a s > 0 , a s = 0 % - - a 3 ~ 0 , a 3 = a 2 - - a t > 0 a n d a x q -a z + a 3 ----- 0 . M o r e o v e r 0 ( 0 ~ 4 - z~ / 2 a n d ~ ( t ) - + -4- o o a s t ~ cx~ a n d t h eo r b i t s l i e o n t h e l e ve l s e t H o = a 2 / 2 . T h e o t h e r h e t e r o c l i n i c o r b i t s a r e o b t a i n e db y a p p r o p r i a t e s ig n c h a n g e s ( c f . F i g u r e 2 ) .

T o a p p ly L e m m a 5 .2 , w e m u s t c o m p u t e~ { H I } ( s h (t ) , O h ( t) ,r h = l , Q h ( t ) + ~ o ) d t .( e o ) = _ H o , ~ - ( ,.~ )

T h e P o i s s o n b r a c k e t i s g i v e n b y8 H o 8 ( H , 1 / 2 ) 8 H o 8 ( H , I1 2)

8 s 8 o 8 o 8 s83/8o= s ( a 3 - f l ( o )) ( bs - r l : c o s ( e + Co )) ~ + - -~ s 8 ~ / 8 o - ~ o ) ) .2 r S f l ( o ( b q - r _ f f o s ( ~ +

( 6 . 3 )U s i n g ( 6 .2 ) , w e s e e t h a t t h e 0 o - i n d e p e n d e n t p a r t o f ( 6 .3 ) is o d d i n t a n d t h e r e f o r ev a n i s h e s i n i n t e g r a t i o n t o y i e l d

r ~ 8 3 1 8 0M ( e o ) = " " } ~ 2 - ~ s ( 3 fl (~ ) - 2 ~ 3 ) c o s ( ~ -t- ~ o ) d t

( 2 )l / ~ /32(o) ~ - ~ (2o% - 3/3(0)) s i n ~ d t s i n P o , ( 6 . 4 )w h e r e w e h a v e a g a i n u s e d t h e o d d / e v e n p r o p e r t i e s o f (6 .2 ) a n d t h e d i f f er e n ti a lr e l a t i o n 8 /3 /8 0 = 2 ~ /~ 2 , a s w e l l a s s e t ti n g r = 1 . E q u a t i o n ( 6 . 4 ) i s e v a l u a t e d i nt h e A p p e n d i x t o y ie l d

t / 1 - b Z ~ c o s e c h ( ~ ) s i n ~ o . ( 6 .5 )

T h i s f u n c t i o n h a s s im p l e z e r o s f o r a l l a t < a 2 < a 3 a n d [ b I < l , a n d t h e p r o o fi n c a s e I is c o m p l e t e .I n c a s e II w e o b t a i n t h e P o i s s o n b r a c k e t f r o m ( 3.1 6 ). A s b e f o r e , th e M e l n i k o v

f u n c t i o n h a s a c o n s t a n t ( ~ o - in d e p e n d e n t) p a r t a n d a p a r t p e r i o d i c in ~o- H o w e v e r ,h e r e t h e c o n s t a n t p a r t d o e s n o t v a n i s h i d e n t ic a l ly . A f t e r u s in g t h e p r o p e r t i e s o f


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C o m p l e x E q u i l ib r i a o f B u c k l e d R o d s 3 3 9t h e u n p e r t u r b e d s o l u t i o n s ( 6 .2 ) a n d i n t e r c h a n g i n g t h e r f l e s o f 062 a n d 0 6a , s o t h a tn o w f l ( a ) = 061 c o s 2 a + 063 s i n 2 a , w e o b t a i n

o o _M ( Q o ) = b ~ - ~ (0 62 - f l ) \ f l - - - ~ s i n a - c o s a 2 f l (a ) s i n a d t

+ 9 )~ / o o ~ ( 0 " ) [ ~ ~ ' ~ s i n o " - - c o s o " s c o s s ( 6 . 6 )

+ ~~ - - - ~ c o s a + s i n s i n ~ d t + _ _ j 2 f l (a ) s i n a c o s ~ d c o s ~ o .T h e s e i n t e g r a ls a r e e v a l u a t e d i n t h e A p p e n d i x t o g i v e

M ( ~ o ) . . . .2 b l / 1 - - b 2 ~0 6 3 r - - 0 6 3 ) ( 0 6 3 - - 0 6 1 )

W e c o n c l u d e t h a t , i f

s e c h ( 063z~ ) COS Po .2 r - 0 6 ) ( 0 6 - 0 6 0( 6 . 7 )

0~]nzb 2 < , ( 6 .8 )062n2 _[ _4 (0 62 _0 63 ) (0 6 3 _ 061) co sh 2 ( 0637'/: )2 ]/(oc2 - 063) (063 - 061)

t h e n M ( ~ o ) h a s s i m p l e z e r o s a n d t h u s t h a t t r a n s v e r s e h o m o c l i n i c o r b i t s e x is t f o re , O s u f fi c ie n t ly s m a l l . T h e p r o o f o f T h e o r e m 4 .1 i s c o m p l e t e . [ ]R e m a r k s o n C a s e H : 061 < 06a " < 0 62 .

I n t h i s c a s e t h e M e l n i k o v f u n c t i o n h a s a c o n s t a n t p a r t a n d , i f l b ] i s s u f f i c ie n t l yc l o s e t o 1 , s o t h a t [b ] < 1 b u t (6 . 8) is v i o l a t e d , t h e n M ( ~ o ) h a s n o z e r o s . T h u s , b y

b- S=1

~ S = - I- A 0

a, 2 2 eF i g . 4 a - c . H y p e r b o l i c f i x ed p o i n t s a n d i n v a r i a n t m a n i f o l d s i n t h e ( a , s ) c r o s s s e c t io n ,r = 1 . ( a) U n p e r t u r b e d p r o b l e m , a l < a 2 < a 3 a n d a l < a a < a 2 . ( b ) P e r t u r b e d p r o b -le m , c~1 < c~2 < c~a a n d a l < a a < a 2 i f b 2 s at i sf ie s ( 6 .8 ) . ( c) A p o s s i b l e s i t u a t i o n i f

a l < a 3 < a 2 a n d b 2 f a i ls t o s a t i sf y (6 . 8 ) s o t h a t M(Oo)h a s n o z e ro .


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340 A. MIELKE & P. HOLMESP r o p o s i t i o n 5 .1 , t h e s t a b le a n d u n s t a b l e m a n i f o l d s t h e t w o p e r t u r b e d f ix e d p o i n t sd o n o t i n t e r se c t . W e o b s e rv e t h a t t h i s d o e s n o t n e c e s sa r il y i m p l y t h a t n o t r a n sv e r s eh o m o c l i n i c o r b i t s e x i s t , s i n c e t h e s t a b l e a n d u n s t a b l e m a n i f o l d s o f a s i n g l e f i x e dp o i n t m i g h t s ti ll i n t e r s e c t, a s i n d i c a t e d i n F i g u r e 4 (c ) . H o w e v e r , t h e p e r t u r b a t i o nc a l c u l a t i o n s g i v e n o i n f o r m a t i o n o n t h i s .

7 . P r o o f o f T h e o r e m 4 . 2A s i s s h o w n i n S e c t i o n 4 t h e s c a le d H a m i l t o n i a n o f a D N - s y m m e t r ic r o d h a st h e f o r m

f i b ( r , S, (r, Q, 1, ~) = /to (r , s , ~) + O It , ( r , s , Q , O) (7.1)+ c ~N -2 7N COS N a + ~(~N --I) ,w h e r e e i s c h o s e n e q u a l t o 1 a n d

1/ Io ( r , s , e) = 89 ~ ~ 2) + ~ ( b s - 7 ; cos r (7 .2)T h e r o t a t i o n a l l y s y m m e t r i c p a r t tS Hs s e v e n i n ~ ( e l (4 .2 )) a n d a l l o w s f o r n o n l i n e a rt e rm s o f l ow er o rde r (d~(6M), 1 =< M ~ N - 2 ) t h an the f i r s t DN -sym me t r i ct e r m 6 N -- 2 ~ N c o s N a .T h e u n p e r t u r b e d e q u a t i o n s a r e

rr= ~ ' 2 s i n 9 ,

~ = - ~ r - ~ - - b 7 + ' ( 7. 3)} = 0 ,

b s7~ = ( ~ - ~ 3 ) s r e r~ ~ c o s O " ~ f ~ ( r ,s , O ) .

H e r e a i s t h e c y c l ic v a r i a b l e . W e r e s t ri c t o u r a t t e n t i o n t o t h e s p e ci a l c a s e s = b4w i t h 0 < b 2 < - - , f o r w h i c h th e p h a s e p o r t r a i t in th e (r , 0 - p l a n e a p p e a r s a s i n0rF i g u r e 5 . T h e l i n e r = b i s d e g e n e r a t e i n t h e ( m , n ) c o o r d i n a t e s y s t e m : f o rr = r - - b 2 = 0 , ( m , n ) i s i n d e p e n d e n t o f 9 ( c f (3.1 b ) ) . Thus the he t e ro c l in i cs o l u t i o n l y i n g o n t h e l e v e l s e t

b 2 - ( r 2 - b 2 ) c o s Q ~ a89 ~(r 2 - b 2) + ~3b 2) + r2 ~ - b 2 + 1 (7 .4)

c o n n e c t in g t h e f ix e d p o i n ts ( r , r ( b, a r cc o s ( - ~ - 1 ) ) i s a h o m o c l in i c o r b itt o t h e s a d d l e p o i n t ( m , n ) = (0 , 0 , b ; 0 , 0 , 1 ) i n t h e o r i g i n a l c o o r d i n a t e s . T h e


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Co m p le x Eq u i l ib r i a o f Bu c k led Ro d s 3414

condi t ion b 2 < - - , n e c e s s a r y f o r t h e s a d d l e t o e x i st , c o r r e s p o n d s t o t h e a n g u l a rOf

m o m e n t u m c o n d i t i o n f o r a " s l o w t o p " i n t h e c la s si ca l L a g r a n g e a n al y si s ( c f .GOLDSTEIN [1980 , C h . 5 ]) .

F ig . 5 . U npe r tu r bed ( r , ~ ) phase p lane fo r equa t io n (7 .3 ) wi th s = b .

W e s h a l l r e q u i r e a n e x p l i c i t e x p r e s s i o n f o r t h e h o m o c l i n i c s o l u t i o n a s a f u n c t i o no f r , o n t h e s e t s = b , a s w e l l a s a n i m p l i c i t r e l a t i o n :

r ( t ) = 7 ( t ) = l / r 2 ( t ) - b Z = y s e c h ( ~ - ~ t ) , (7 .5 a )

4w h e r e 7 2 = -- -- b 2 > 0.T o a p p l y th e M e l n i k o v m e t h o d w e m u s t c o m p u t e t h e P o i ss o n b r a c k e t

H 1 c~Ho ~H o H i c0O 0H o c~Ho H 1H o , - - f f - - e e 0 2 ~ - ~ ~ r ~ 2 ~ r " ( 7 . 6 )F r o m ( 7 . 3 ) a n d ( 7 . 5 b ) w e s e e t h a t , o n t h e h o m o c l i n i c o r b i t , O ( r , s ---- b , ~ ) i sc o n s t a n t a n d t a k e s t h e v a l u e

(77,w h ic h i s n o n z e r o s in c e o% > Of.

T h e p e r t u r b a t i o n H a m i l t o n i a n H 1 d i v id e s i n t o t w o p a r t s : d H s a n d0 H o 0 H o 0Y2c 6 N- 2 ~N cos N a . S ince Ho , Hs an d ~2 a re e ve n in Q, ~ , ~---~- an d ~ a re o d d

i n ~ a n d { H o , ~ - L } i s a l so o d d i n ~. F r o m t h i s f a c t a n d t h e e v e n n e ss o f r , 7 , a n d ri n t w e d e d u c e t h a t t h i s p a r t o f t h e P o i s s o n b r a c k e t i s o d d i n t a n d s o d o e s n o tc o n t r i b u t e t o t h e M e l n i k o v i n te g ra l . T h u s w e n e e d o n l y c o m p u t e t h e p a r t i n v o lv i n g

2r 2 = - - (1 + c o s ~ ) , (7 .5 b )Of


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3 4 2 A. MIELKE & P. HOLMESclaN--2 ~,N COS N a . U s i n g ( 7 . 6 a , b ) , w e o b t a i n

~ H o ~ F ~ ~ H o F 2~ r r ' cgQ r 2 si n ~ ,~ ( H 1 / t 2 ) c b 7Nc o s N a , ~Q ~Q2r s in r c os N a . (7 .8 )~r -- f22r}

T h e M e l n i k o v i n t eg r a l t h e r e f o r e r e d u c es t oM (~ o ) = - T s in ~ c o s (N(~ + Co )) d t , (7 .9 )

w h e r e a ( t ) = D t , s in c e s i s c o n s t a n t .U s i n g ( 7 . 8) a n d t h e f a c t t h a t r , 7 a r e e v e n i n t w h i l e s i n 0 is o d d a n d t" = ( 7 2 / r2 )s i n ~ , w e m a y r e w r i t e ( 7 .9 ) a s f o l l o w s :

( ~ s i n ( N ( f 2 t ) ) N r - r N - 2 r d t ) s i n ( N a o ) .M ( ~ o ) = cI n t e g r a t i n g b y p a r t s a n d s u b s t i t u ti n g f r o m ( 7 .5 ) , w e o b t a i n

M ( ~ o ) = e ( / Z - N c o s ( N ~ t ) - r N ( t ) d t ) s i n ( N a o )= - N e y N ( s_ - 2 N e ' / V - l ( s- ( 2 N $ 2 ~ l d z t s in (Ntro) .

2NS2W r i t i n g t h e i n t e g r a l o f ( 7 .1 0 ) a s IN ( to ) , to - - - -oWb y p a r t s t w i c e w e d e d u c e t h e r e c u r r e n c e r e l a t i o n

to2 _~ N 2IN+2(to) -- N ( N + 1) I N ( t o ) '

b N ( 1 - ~ 2 ) , b y i n te g ra t in gVN --> 0. (7 .11 )

T h i s r e l a t io n , t o g e t h e r w i t h t h e e v a l u a t i o n s o f I~ a n d I 2 b y t h e c a l c u lu s o f re s id u e s ,n a m e l y

/1 ( to ) = z~ s e c h ( - ~ ) ,(7 .12)

I z ( t o ) = m o c os e ch ( - ~ ) ,g u a r a n t e e s t h a t I N ( to ) ~ = 0 f o r a l l N ~ 1 a n d or y 4 : 0 . W e c o n c l u d e t h a t t h eM e l n i k o v f u n c t i o n ( 7. 10 ) h a s s i m p l e z e r o e s u n d e r t h e h y p o t h e s e s o f t h e t h e o r e m .A p p l y i n g T h e o r e m 5 .3 c o m p l e t e s t h e p r o o f . [ ]R e m a r k . W e h a v e a s s u m e d N ~ 3 , s o t h a t t h e e x p l ic i t a - d e p e n d e n t te r m i n t h eH a m i l t o n i a n a p p e a r s a t h i g h e r o r d e r . I n t h e c a se N - -- - 2 t h e r e l ev a n t t e r m i sc~ 2 c o s 2 a, w h i c h o c c u r s a t t h e s a m e o r d e r a s t h e r o t a t i o n a l l y s y m m e t r i c p a r t H o .


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3 4 4 A . M I E L K E P . H O L M E ST h e M e l n i k o v i n t e g r a l t h e n y i e ld s

~ S H o 8 ( H a l f2 ) O Ho 8 ( H , / Q ) ~M ( 9 ~ = / \ 8 s 8 a 8 a -~ s " ] d t= f {s(o,~ - f l) [bs - ~ / ~ t / 1 - s 2 c o s ( 9 + 9 o )1R

b I/1 - - b 2+ 8 9 ~ ) fl + r s ~ 8tr6qTS i n c e s , f l c o s 9 a r e e v e n i n t a n d s i n 9 , ~ a a r e o d d t h i s i n t e g r a l s i m p l i f ie s , a f t e r

u s e o f (A . 4 ) , t oM (0 o ) ----- b" 0 + ] / 1 - b 2 I * s i n ~ o ,

s ~-1 " Z l /1 - s 2 f12 ( 2 o~ a - 3 f l ) s i n 9 d r ,

a s in e q u a t i o n ( 6 .4 ) . E x p r e s s i n g f l i n t e r m s o f s v i a ( A . 2 b ) , w e h a v es* = f g ( s ) ~ s i n 9 d t ,

Rw h e r e

a n dg ( s ) - s ~/1 - s 2( a 2 - aas2) 2 ( 2 a 3 ( 1 - s 2 ) - 3 ( c% - a a s 2 ) )

a ( s ) = f g ( O d s - - ( 1 - s : ) 3 / 2 t / 1 - s 2~ ,~ - ~ , ~H e n c e , u p o n i n t e g r a t i o n b y p a r t s , w e h a v e

f g(s) "ss i n Q d t = G(s ) s i n Q - f G(s) ~ c o s 9 d tI / 1 - s 2= f l s i n g + f t / 1 - s 2 c o s g d t

s i n 9 - - s i n 9 + - - S 2 ( t ) c o s o~2t d t.0r 2T h e l a s t e q u a l i t y i s a c o n s e q u e n c e o f ( A . 5 a ) . N o w

I * = [ ( 1 - s 2 )- _ -

( A . 6 )

) i a 3 V1/1 _ s Z s i n Q + - - S 2 ( t ) c o s ~ 2 t d r ,2" --oo 0 r - -


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Com plex Equ i li b r ia o f Buck led Rods 345s o t h a t t h e b o u n d a r y t e rm s v a n i s h a n d , l e tt in g ~ = ] / a ~ a 3 t , w e a r e l e f t w h i t h

1 ] / ( ~x2 )I * = - - a3 f s e c h 2 ( 'r) c o s ~ v d roc2 r- -a 2t t \g al as /( . ~ 2 ] ( A . 7 )1/_----~la~ cosech \ 2 l / a ~ a ] ,

f r o m w h i c h w e o b t a i n ( 6.5 ). T h e f i na l i n t e g r a l i s e v a l u a t e d b y t h e m e t h o d o f r e si -d u e s .F o r t he case 0 < 0r < o~a < o~2, as i nd i ca t ed in Sec t ion 4 , we u se the ( r , s , a , 0 "c o o r d i n a t e s f o r t h e v e c t o r s ( m ~ , m a , m 2 ) r a n d ( n l , h a , n 2 ) r r a t h e r t h a n f o r ma n d n . T h u s , t h e u n p e r t u r b e d s o l u t i o n s a r e g i v e n b y t h e f o r m u l a e ( 6 . 2 ) , ( A . 2 )a n d ( A . 3 ) , a s a b o v e , b u t w i t h o~2 a n d or i n t e r c h a n g e d a n d t h e h e t e r o c l i n i c o r b i t sn o w c o n n e c t th e p o i n t s m = (0 , 0 , 4 -1 ) . T h e p e r t u r b a t i o n H a m i l t o n i a n , H 1 ,h o w e v e r i s q u i t e d i f f e r e n t . F r o m ( 3 . 1 6 ) w e h a v e

/ 4 0 = ~ ( ( ~ - s 2 ) ~ ( ~ ) + ~ 2 s 2 ) ,/ / i b r s i n a r

- - r 3 f l r a f i ( s s i n a c ~ + r c o s a s i n o )Dw h e r e n o w f l ( a ) = oq co s 2 a + 0% sin 2 a .

T h e M e l n i k o v f u n c t i o n i sM ( 9 ~ : 9 ~ 1, 8 s ~ a ~ a ~ s "} d t .

W i t h r ---- 1 a n d a f t e r o m i t t i n g th e o d d t e r m s i n th e i n t e g r a n d , w e a r e l e f t w i t hM ( 9 o ) ---- b i t + I /1 - b 2 ( /2 - h ) cos Co, (A.8 )

w h e r e 1 1, 12 a n d I a a r e t h e i n t e g r a l s d e f i n e d i m p l i c i t l y b y c o m p a r i s o n o f ( A . 8 )and (6 .6 ) . Fo r t he f i r s t , we have

~ = f r - - - 2 7 ( ~ - f i( ~) ) ~ N - - 3 - / 2 ~ (~ ) ~ s i n ~ d r. ( a . 9 )U s i n g b = s ( f i - o~2) f ro m (6 .1 ) (wi th a3 4 - + 0 ( ' 2 ) a n d f l ' ( a ) = 2k /72 , we m ay re -w r i t e ( A . 9 ) a n d o b t a i n

n/2 e~ .11 f - - d ( - s i n a i / - _ssm z= t 1 - s z ~ a \ ~ ] d a - _ .] f l ( a ) ] / ~ - s 2 j d t- hi2

_ _ - - z - 7 ( - s i n , q hi2 hi2,~ /2 _ I " . s . s i n g _ d s f s s i n a d s- ~ 1 2 - ~ 2 f l ( a ) t /1 - s 2 d a d a 4 - _ = 1 2 f l( a ) - -~ - - ~ _ s 2 a a d ~

2 , (A. IO )o~ 3

s i n c e t h e t w o l a s t i n t e g r a l s c a n c e l o n e a n o t h e r .


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346 A. MIELKE~; P. HOLMESI n t h e 9 o - d e p e n d e n t p a r t o f (A . 8 ) w e h a v e

oo I s d ( s i n a i d ( c o s a l l1 2 = _ oo s ( ~ 2 - f l( a ) ) c o s 9 ~ - - f l ( a ) ] s i n g ~ - - ~ - ] ] d r , ( A . 1 1 a )

f ( 1 - s 2) eft/ 3 = 2 ~ 5 e ~ s i n ~ c o s 9 d r . ( A . 1 1 b )- -oo

U s i n g s( f l - c~2) = b a g a i n w e h a v e , i n t e r m s o f d a ,~ /2 [ d ( s i na ' \ d ( c o s a ] ]I 2 - - _f~/2 s c o s q ~ - f l- -~ -~ )+ s i n 9 7 / - ~ a - - - ~ ] j d a ,

o r , a f t e r i n t e g r a t i o n b y p a r t s a n d t r a n s f o r m a t i o n b a c k t o d t :hi2 r ]/2 f [ s i n d d d a= - ~/2 t a ~ (s co s 9) + co s a ~aa (si n 9) fl-(-~')

[ d d 9) ] d t: - ~ f s i n a - d - [ ( s c o s g ) + c o s a - ~ ( s i n j f l ~ ) . ( A . 1 2 )N o t e t h a t , i n i n te g r a t i o n b y p ar ts , t h e b o u n d a r y t e r m s v a n i s h si n ce s a n d c o s a - + 0a s a - + ~ z z t /2 . I n 13 w e u s e (A . 4 ) to o b t a i n

I 3 = - o o f b S i n a c ~ (A .1 3 )T h u s , f r o m ( A . 1 2 ) a n d ( A . 1 3 ), w e h a v e , a f t e r c a n c e l l a t i o n a n d u s i n g ~ = - - f l ( a ) :

12 - 13 = f [s s in a s i n 9 - cos a cos 9] d t . ( A . 1 4 )- -oo

F i n a l l y , w e u s e t h e r e l a t io n s ( A .2 ) ( w i t h a 2 ~ a 3 , s o t h a t a ~ --> - a l , a 2 --> - a 3 ,a 3 - + - a 2 ) a n d t h e f a c t t h a t S 2 + T 2 - ~ 1 t o r e d u c e ( A .1 4 ) to t h e f o r mI 2 - I 3 = - f S ( t ) c o s o r d t

_ 1~ala2

Vala2

- - f s e c h T c o s d T-oo \t/a---~-a%- - s e c h \ ~ / .

T h u s f r o m ( A . 8 ) , ( A . 1 0 ) a n d ( A . 1 5 ) , w e h a v e

s e e n / ~ ! c os 9 o,M ( 9 ~ = c~3 ] / a l a 2 \ 2 1 / a l a 2 ]w h i c h g i v e s ( 6 . 7 ) .

( h . 1 5 )

( A . 1 6 )


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Complex Equilibria of Buckled Rods 347Acknowledgment. The research reported here was supported by ARO under grantDAAG 29-85-C-0018 (Mathematical Sciences Institute), NSF under MSM 84-02069

and AFOSR under 84-0051.

References

R. ABRAHAM~; J. E. MARSDEN [1978] Foundations of Mechanics, 2nd edition, Benjamin/Cummings, Reading, MA.S. S. ANTMAN [1984] Large lateral buckling of nonlinearly elastic beams, Arch. RationalMech. Anal. 84, 293-305.S. S. ANTMAN & C. S. KENNEY [1981] Large buckled states of nonlinearly elastic rodsunder torsion, thrust and gravity, Arch. Rational Mech. Anal. 76, 289-338.V. I. ARNOLD [1964] Instability of dynamical systems with several degrees of freedom.Soy. Math. Dokl. 5, 581-585.V. I. ARNOLD [1978] Mathematical Methods of Classical Mechanics Springer Verlag,N.Y., Heidelberg, Berlin (Russian original, Moscow, 1974).G. D. BmKnOrF [1927] Dynamical Systems, A.M.S. Publications, Providence, R.I.E. BUZANO, G. GEYMONAT, & T. POSTON [1985] Post-buckling behavior of a nonlinearlyhyperelastic thin rod with cross section invariant under the dihedral group D n, Arch.Rational Mech. Anal. 89, 307-388.L. EULER [1744] Additamentum I de Curvis Elasticis, Methodus Inveniendi Lineas CurvasMaximi Minimivi Proprietate Gaudentes, Lausannae in Opera Omnia 1 24, Fiissli,Ziirich [1960], pp. 231-297.B. O. GREENSPAN P. J. HOLMES [1983] Homoclinic orbits, subharmonics, and globalbifurcations in forced oscillations, in Nonlinear Dynamics and Turbulence (editors:B. Barenblatt, G. Iooss & D. D. Joseph) Pitman, London. Chapter 10, pp. 172-214.H. GOLDSTEIN [1980] Classical Mechanics, Addison-Wesley, Reading, MA.J. GUCKENHEIMER& P. J. HOLMES [1983] Nonlinear Oscillations, Dynamical Systems andBifurcations of Vector Fields, Springer Verlag, New York (corrected second printing,1986).P. J. HOLMES & J.E. MARSD~N [1982] Horseshoes in perturbations of Hamiltoniansystems with two degrees of freedom, Comm. Math. Phys. 82, 523-544.P. J. HOLMES & J. E. MARSDEN [1983] Horseshoes and Arnold diffusion for Hamiltoniansystem on Lie groups, Indiana University Math. J. 32, pp. 273-310.G. KIRCHHOFr [1859] (3ber das Gleichgewicht und die Bewegung eines unendlich diJnnenelastischen Stabes, Journal fiir Mathematik (Crelle) 56, 285-313.P. S. KRISHNAPRASAD, . E. MARSDEN & J. C. SIMO [1986], The Hamiltonian structure ofnonlinear elasticity: the convective representation of solids, rods and plates, (preprint).J. LARMOR[1884] On the direct applicat ion of the principle of least action to the dynamicsof solid and fluid systems, and analogous elastic problems, Proc. London Math. So-ciety 15, 170-184.A. E. H. LOVE [1927] A Treatise on the Mathematical Theory o f Elasticity CambridgeUniversity Press (4th edition).V. K. MELNIKOV [1963] On the stability of the center for time period perturbations,Trans. Moscow Math. Soc. 12, 1-57.A. MIELKE [1987] Saint-Venant's problem and semi-inverse solutions in nonlinear ela-sticity, MSI Tech. Rep. '87-25, Arch. Rational Mech. Anal., to appear.J. MOSER [1973] Stable and Random Motions in Dynamical Systems, Princeton UniversityPress, Princeton, N.J.


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3 4 8 A . MIELKE & P. HOLMESS. SMALE [1963] D i f feom orp hism s wi th m an y p er iod ic p o in ts , in Dif feren t ia l and C ombi-

national Topology, ed . S . S . CAIRNS, pp . 63-80 , Pr ince ton Un ivers i ty Press , Pr ince-t o n , N . J .S . SMALE [1967] Di f fe ren t iab le D yna m ica l Sys tem s , Bul l . Amer . Math . Soc . 73, 747-817.

E. T. WHITTAKER [1937] A trea ti se on the Analy t ica l Dynam ics o f Par t ic les and Rig idBodies ( 4 t h e d i t i o n ) C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e .M a t h e m a t i s c h e s I n s t i t u t AUnivers i t / i t S tu t tgar tF e d e ra l R e p u b li c o f G e r m a n y

a n dMathemat ica l Sc iences Ins t i tu te andD e p a r t m e n t s o f T h e o re t ic a l & A p p l ie d M e c h a n ic sa n d M a t h e m a t i c sC o r n e l l U n i v e r s it yI th a c

Alexander Mielke and Philip Holmes- Spatially Complex Equilibria of Buckled Rods

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