Integrable and Nonintegrable

7/30/2019 Integrable and Nonintegrable

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I N T E G R A B L E A N D N O N I N T E G R A B L E H A M I L T O N I A N S Y S T E M S

I a n P e r c i va lSchool of Mathematical SciencesQueen Mary CollegeU n i ve r s i t y o f L ondon ,Mile End RoadLo nd on E1 4NS. U.K.

1: Moder n Dynamics and I ts Applicat i onsI n t r odu c t i o nTradi t iona l ly Hami l tonian sys t ems w i th a f in i t e n umbe r of degr ees of f r ee dom havebeen d iv ided in to those w i th f ew deg rees of f r eedom which were supp osed to exhib i ts ome k i nd o f r e gu l a r o r de r e d m o t ions a nd t hos e w i t h l a r ge num be r s o f de g r e e s o ff r eedom for which the methods of s ta t i s t i ca l mechani cs should be use d .

The la s t f ew deca des ha ve see n a comple te chan ge of v iew [1]. This change of v iewa f f e c t s a l m os t al l t he p r a c t i c a l a pp l i c a t i ons , pa r t i c u l a r l y i n m a t he m a ti c a l phys i c s ,which has b een domina ted for many decad es by l in ea r mathemat ics , coming f romq u a n t u m t h e or y.

T h e m o t i o n o f a H a m il t o n i a n s y s t e m i s u s u a l l y n e i th e r c o m p l e t e l y r e g u l a r n o rp r o p e r l y d e s c r i b e d b y t h e m e t h o d s o f s ta ti st ic al m e c h a n i c s . It e xh ib it s b o t h r e g u l a ra nd irreg ular o r chaotic mot ion f or different initial conditions, a n d the transitionbe tw ee n the tw o ty pe s of motion, as t he initial condit ions are varied, is complicated,subtle and beautiful.

The na ture of the r egu la r mot ion in a sys te m of m degr ees of f r eedom i s the samea s t ha t o f t he t r a d i t i ona l i n t e g r a b l e s y s t e m s ; w he n bounde d i t i s qua s i pe r i od i ca lmos t ever ywhe re , w i th a d i sc r e te se t o f m f r eq uenc ies , t%, tog e the r w i th the i rin t eger l inea r combina t ions . For cons e rva t iv e sys tems , when the Hamil tonian i sinde pend ent o f the t ime , the r e gula r mot ion for a g iven ini t i a l condi t ion is conf inedto an m-dimen s iona l r eg ion in the 2m-dimens ion a l phas e spac e .

By cont r as t~ the na t ure of chaot ic mot ion i s s t il l no t fu l ly unde r s to od . I t i s uns t ab l ei n a s t r ong e xpone n t i a l s e ns e . F o r a c ons e r va t i ve s y s t e m it u sua l l y c a nno t bec on f i ne d t o a ny s m oo t h r e g i on o f d i m e ns i on l e s s t ha n 2m- 1, t he c o n f i ne m e n t r e qu i r e db y e n e rg y c o n s e r v a t i o n .


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B ut it doe s no t no r m a ll y oc c upy t he w ho l e o f a n e ne r gy s he l l a s r e qu i r e d by t h ee r god i c p r i nc i p l e o f t r a d i t i ona l s t a t i s t i c a l m ec ha n i c s . F a r aw a y f r om r e gu l a r r e g i onso f pha s e s pa c e , t he c ha o t i c m ot i on r e s e m b l e s a s i mp l e d i f f u s i on p r oc e s s , bu t i n t hene i ghbo urho od of r egu la r r eg i ons i t does not . Chaot ic motion i s common forc ons e r va t i ve s y s t e m s o f 2 de g r e e s o f f r e e dom .

Now cons ide r how th i s ch ange of v iew a f f ec t s some spec i f ic appl ica t i ons of dynamicsa nd a l so t he r e l a t i on be t w e e n dyna m i ca l t he o r y a nd a pp l i c a t i ons .

C l a s s i ca l S t a t i s t i c a l M e c h a n i c s

T he f ounda t i ons o f s t a t i s t i c a l m e c ha n i c s r e qu i r e t he num b e r o f de g r e e s o f f r e e dom totend to in f in i ty . Rea l sys t ems have f in i te numb er s of degr ees of f r eedom. Thenum be r s m a y be ve r y l a r ge bu t t h e m e t h ods o f s t a t i s t i c a l m e c ha n i c s a r e a pp l i e de ve n w he n t he num be r i s qu i t e s m a l l , a s f o r t he E dd i ng t on t he o r y o f s t a r s i n t hesmoothed grav i t a t io na l f i e ld of ga lax ies~ or the r ed i s t r i bu t ion of modes of v ib ra t i onin molecules .

A mode l o f the l a t t e r was used in a key nume r ica l exper i ment o f Fermi, Pas ta andUlam [2] (1955) . A one dimen siona l dynam ical sy s te m of 64 par t ic les with n onl ine ari n t e r a c t i on w a s s t ud i e d by c om pu t e r .

The r esu l t s were Fou r ie r a na ly sed a nd p lo t ted as a func t io n of t ime for 30,000 or80,000 bas ic t ime uni t s . They showed very l i t t l e , i f any , t e nde ncy towa rdse qu i pa r t i t i on of e ne r gy a m ong t he de g r e e s o f f r e e dom a s w ou l d be r e qu i r e d bys ta t i s t i ca l mechanics .

T hus , t he r e i s a ppa r e n t l y r e gu l a r m o t ion i n a s y s t e m wi t h l a r ge num be r s o f de g r e e so f f r e e dom w he r e none w a s e xpe c t e d . T h i s r e s u l t i s r e l a t e d t o t he t he o r y o fsol i tons , as wel l as to chaot ic motion.

S i nc e t he n i t h a s - be e n pos s i b l e t o i nve s t i g a t e t he t r a n s i t i on f r om r e gu l a r t o c ha o t i cm ot ion i n s uc h c ha i n s a nd t he s e ha ve be e n s t ud i e d pa r t i c u l a r l y by F o r d , C a sa t i a ndt he i r c o l l a bo r a t o r s [ 2]. T he y ha ve f ound t ha t w he n t he mo t ion i s s u f f i c i e n t l ychaot ic , the t he rmal condu c t i v i ty i s normal to a good approximat ion , so tha t the r ea r e now s ys t e m s f o r w h i c h t he num be r o f pa r t i c l e s i s s u f f i c i e n t l y s ma ll f o r t hedynamics to be computab le , ye t an e f f ec t ive compar i son can be made wi th the r e su l t sof s ta t i s t i ca l mechanics .


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H owe ver , we a r e no t a b l e t o p r e d i c t a na l y t i c a l l y w he t he r a pa r t i c u l a r r e s u l t o fs ta t i s t i ca l mec hanics wi ll be va l id for a g iven sys t em of a f in i t e numb er of degre esof f r eedom, except fo r g ross ly s impl i f i ed mode ls .

Confinement of Part ic les

C ha r ge d pa r t i c l e s m a y be c on f i ne d ( by e l e c t r om a gne t i c f i el d s ) f o r l ong pe r i od s o ft ime in f in i te r eg ions of space .

T h i s h a p p e n s n a t u ra l l y in t h e r ad ia ti on b e l ts w h e r e t h e p ar ti cl es a r e t r a p p e d b y t h eE a r th ' s m a g n e t i c f i e l d. T h i s i s a p p r o x i m a t e d w el l b y t h e t r a p p i n g o f a c h a r g e dp ar ti cl e b y a m a g n e t i c d ip ol e, a n o n i n t e g r a b l e H a m i l to n i a n s y s t e m w i t h b o t h r e g u l a ran d chaotic motion [3].

Bu t the mo re well k n o w n exa mple s of con fin eme nt of particles are artificial.F u n d a m e n t a l p ar ti cl es a re c o n f i n e d b y e l e c tr o m a gn e t i c f i e l d s i n v a c u u m s y s t em s ,a c c e le r a t e d a n d s o m e t i m e s s t o r e d f o r m a n y h o u r s , s o t h at t h e y a r e a v ai la bl e f o r h i g hen er gy collisions to aid the basic studies of particle interactions. Ions are c onfi nedi n p l a s m a s a t h i g h t e m p e r a t u r e a n d d e n s i t y i n o r d e r t o p r e p a r e f or t h e e v e n t u a lp r o d u c t i o n o f u s e f ul e n e r g y b y t h e r m o n u c l e a r r e a c ti o n s - c on tr ol le d t h e r m o n u c l e a rreactions.

I n b o t h c a se s t h e a p p a r a t u s is v e r y e x p e n s i v e a n d f or m a x i m u m v a l u e t h eexpe rim ente rs wa nt to confine t he particles for as long as possible at as hig h adensity as possible.

T he h i gh de ns i t i e s p r oduc e t he i r ow n e l e c t r om a gn e t i c f i e l d s . I n be a m - be a mi n t e r a c t i o n de v i c e s t he e l e c t r om a gne t i c f i el d o f one p a r t i c l e be am m od i fi e s t hepa r t i c l e o r b i t of t he o t he r . F o r p l a sm a c on t a i nm e n t de v i c e s li ke T okam a ks t hee l e c t r on a nd i on c u r r e n t s p r oduc e s t r ong m a gne t i c f i e l d s t ha t a r e d i f f i c u l t t oc on t ro l . T he s e e l e c t r om a gne t i c f i e l d s r e a c t ba c k on t he pa r t i c l e s . A s t he de ns i t yi nc r e a s e s t he e qua t i ons o f mo ti on o f t he pa r t i c l e s c ha nge s o t ha t t he pa r t i c l e sbecome more d i f f i cu l t to conf ine .


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O f t e n t h e r e a s o n f o r t h i s i s t h a t t h e p a r t i c l e m o t i o n b e c o m e s c h a o t i c o v e r as i g n f i c a n t f r a c t i o n o f t h e p h a s e s p a c e . F o r t h i s r e a s o n t h e t r a n s i t i o n f r o m r e g u l a rt o c h a o t i c m o t i o n i s p a r t i c u l a r l y i m p o r t a n t f o r c o n f i n e m e n t , a n d m a n y o f t h e o r i g i n a ls t u d i e s o f t h i s t r a n s i t i o n w e r e s t i m u l a t e d b y p r o b l e m s o f c o n f i n e m e n t [ 4 ] .

I s h a l l b e d i s c u s s i n g t h e p r o b l e m o f c o n f i n e m e n t i n m o r e d e t a i l l at er . It i s o n l yPartially unde rst ood.

S o l a r Sys tem

C l a s s i c a l d y n a m i c s o r i g i n a t e d w i t h t h e S o la r S y s t e m . T h e a p p a r e n t l y r e g u l a r m o t i o no f t h e S u n a n d t h e p l a n e t s i n t h e h e a v e n s l e d d i r e c t l y t o t h e d i s c o v e r y o f t h e b a s i cl a w s o f m o t i o n .

We n o w k n o w that it consists of the Sun , the planets, their satellites, their rings,t h e a s t e r oi d s a n d t h e c o m e t s , all o f t h e m s u b j e c t t o t h e l a w s o f d y n a m i c s .

H o w r e g u l a r i s t h e m o t i o n ?

A t o n e e x t r e m e t h e m o t i o n o f t h e c o m e t s a p p e a r s t o b e i r r e g u la r a n d a t t h e o t h e rt h e m o t i o n o f t h e p l a n e t s a p p e a r s t o b e p e r f e c t l y r e g u l a r . B u t i s i t ? I n o n e s e n s ei t c a n n o t b e t b e c a u s e t h e i r r e g u l a r m o t i o n o f a n y b o d y i n t h e S o l a r S y s t e m i m p l i e st h e i r r e g u l a r m o t i o n o f a n y o t h e r . B u t t h e m a s s e s o f t h e c o m e t s a r e s o s m a l l t h a tt h e y h a v e a n e g l i g i b l e e f f e c t o n t h e p l a n e t s o v e r m i l l i o n s o f y e a r s a n d a l s o , t o af ir st a p p r o x i m a t i on , o n e c a n s u p p o s e t ha t t h e m a s s e s o f t h e sa te ll it es a r e c o m b i n e dwith those of the planets. Asteroi ds do affect the inner planets.

I s t h e m o t io n o f t h e S o l a r S y s t e m r e g u l a r i n t h i s r e s t r i c t e d s e n s e o f t h e d y n a m i c so f t h e S u n a n d 9 p l a n e t s ? W e s t i l l d o n o t k n o w . A c r u c i a l t h e o r e m i n t h i sc o n n e c t i o n is t h e f a m o u s K o i m o g o r o v - A r n o l d - M o s e r ( K A M ) t h e o r em , w h i c h s h o w s t h a tf or s uf fi ci en tl y sm al l a n d s m o o t h p e r t u r b a t i o n s o f a n i n t e g r a b l e H a m i l to n i a n S y s t e m ,m o s t o f t h e p h a s e s p a c e is c ~ c u p i e d b y r e g u l a r or bi ts . If w e s u p p o s e t h e S u n t oh a v e i n f i n i t e m a s s a n d n e g l e c t t h e i n t e r a c t i o n s b e t w e e n t h e p l a n e t s , t h e n t h e S o l a rS y s t e m i s i n t e g r a b l e , a s N e w t o n d i s c o v e r e d . A l t h o u g h t h e p e r t u r b a t i o n c a u s e d b yt h e i n t e r a c t i o n b e t w e e n t h e p l a n e t s i s s ma ll~ i t i s n o t s u f f i c i e n t l y s m a l l f o r t h e K AMt h e o r y t o a p p l y .


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In f ac t Hdnon [5] showed Arnol d ' s p roof on ly appl ies i f the pe r turb a t io n i s l e s s tha n10 333 a nd Mose r's if it is less tha n 10 48, in appropri ate units. Th e latter is lesst h a n t h e g r a v it a t io n p e r t u r b a t i o n o f a f o ot ba ll i n S p a i n b y t h e m o t i o n o f a b a c t e r i u min Australia!

The KAM proo f s were a v i ta l con t r ib u t io n to dynamics b ecaus e th ey showed tha tr e gu l a r m ot i on i s no t e f f e c t i ve l y r e s t r i c t e d t o i n t e g r a b l e s y s t e m s , bu t num e r i ca l l yt he y a r e no t ye t of p r a c t i c a l va l ue . T he r e a r e c ons i de r a b l e e f f o r t s by Ga l la vot t i,Russmann, Herman and o the r s to improve them [6].

B e t w ee n t he c om e ts a nd t he p l a ne t s a r e t he a s t e r o i d s a nd t he s a t e l l i te s . T he r e iss t ro ng ev i dence , pa r t i cu la r l y f rom the work of Wisdom and o the r s [7] tha t a l th oughthe i r mot ion i s ma in ly r egu la r , tha t i r r egul a r mot ion p la ys a ro le in the format ion ofthe K i rkwood gaps in the as te ro ids .

Semiclassica] Me ch an ic s

T h e u s e o f c l a s s i c a l d y n a m i c s f o r t h e s t u d y o f mo le cu le s, a t o m s , n u c le i a n df u n d a m e n t a l p ar ti cl es is c o n t r a v er s i a l b e c a u s e q u a n t u m e ff ec ts c a n b e i m p o r t a n t o re v e n d o m i n a n t .

O f co u rs e , th e p l a ne t s a r e al so s u p p o s e d t o o b e y t h e l a w s of q u a n t u m m e c h a ni c s ,b u t t h e s p e ci f i c q u a n t u m e ff ec ts a r e s o sm al l as t o b e u n o b s e r v a b l e a n d n o o n et a k e s s e ri o u sl y a q u a n t u m t h e o r y o f t h e S o l ar S y s t e m .

T h e m o t i o n o f e l e c tr o n s in a t o m s a n d m o l e c u l e s is n o r m a l l y a t t h e o t h e r e x t r e me .T h e s ol ut io ns o f t h e c l a ss i c al e q u a t i o n s o f m o t i o n f o r a s y s t e m w i t h m o r e t h a n o n ee l e ct r o n a p p e a r t o h a v e l it tle r el at io n w i t h t h e o b s e r v e d b e h a v i o u r o f t h e s y s t e m s ,I t w a s t hi s d i f f er e n c e t h a t l e d to t h e d i s c o v e r y o f q u a n t u m m e c h a n i c s .

Th e vibrations an d rotations of mole cul es are particularly interesting be ca us e t heylie b e t w e e n t h es e t wo e x t re m e s [ 8 ] . A c c o r d i n g t o t h e B o r n - O p p e n h e i m e ra p p r o x i m at i o n , t h e n uc le i o f t h e a t o m s m o v e i n a p o te nt ia l d e t e r m i n e d b y t h eelectronic motion. Th e vibrations a nd rotational propertie s of the molecule are the nd e t e r m i n e d b y t h e e q u a t i o n o f m o t i o n o f t h e a t o m s i n t hi s p ot en ti al . T h i sapp rox ima tio n is excellent.

C l as si c al dyna m i c s c a n be u s e d ve r y e f f e c t i ve l y to ga i n i n s i gh t i n t o t he s e p r ope r t i e so f m o le c ul e s, e ve n t hough qua n t um e f f e c t s a r e c l e a r l y obs e r va b l e .


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T he r e l a t i on be t w e e n c l a s s i c a l a nd qua n t um m e c ha n i c s a ppe a r s on t w o l e ve l s:formula t ion and so lu t ion .

O n e r el at io n a p p e a r s i n t h e p r o b l e m o f q ua nt iz a ti on : h o w c a n o n e o bt a i n t h ef o r m u l at i o n o f a q u a n t a l s y s t e m f r o m t h e f o rm u l a t i o n o f t he c o r r e s p o n d i n g c la s si ca ls y s t e m ? A n e x a m p l e is t h e d e r iv a t i o n o f t h e w a v e e q u a t i o n f r o m t h e c la ss ic alllamiltonian.

T h e o t h e r r el at i on a p p e a r s i n t h e p r o b l e m o f se mi cl a ss ic al m e c h a n i c s : w h a t i s t h er el at io n o f t h e s ta t es o r m o t i o n o f q u a n t a l s y s t e m s t o t h e m o t i o n o f t h ec o r r e s p o n d i n g c l a ss i ca l s y s t e m ? A n e x a m p l e i s t h e a p p r o x i m a t e d e t e r mi n a t i on o fe n e r g y l e v e l s f r o m c la ss ic a l o r b i t s, a s i n t h e ol d q u a n t u m t h e o ry .

The second leve l may not appe ar to be as fund amen ta l a s the f i r s t , bu t i t i s s t i l li m por t a n t. T he c l a s s i c a l e qua t i o ns m a y be s o l ub l e w he n t he qua n t a l e qua t i ons a r enot . We l ive in a macroscopic c las s ica l wor ld and any i ns i gh t in to the r e la t ionbe t w e e n t h i s wor l d a nd t he m i c r o s c op ic qua n t a l w orl d he l p s u s t o unde r s t a nd t hela t t e r .

T he s e m i c l as s i c a l m e c ha n i c s of c ons e r va t i ve s y s t e m s o f one de g r e e o f f r e e dom i sr e a s o n a b l y w el l u n d e r s t o o d . T h e r e a r e f o r m a l s em ic la ss i ca l e x p a n s i o n s f or' v ib ra ti on a l a n d r o t a t i on a l m o t i o n a n d e x p r e s s i o n s f o r b a r r i e r p e n e t r a t i o n [ 9 ] . T h e r ea r e u n i f o r m a p p r o x i m a t i o n s t h at b r i d g e t h e g a p s b e t w e e n t h e s e c a s e s [ 1 0 ] a n d f o ranalytic potentials there is ev en a th eo ry relating exact qua ntal solution to com ple xc l as si ca l o r b i t s [ 1 1 ] , t ha t h a s b e e n c a r r i e d t h r o u g h e x p l i ci t l y f o r t h e q u a r t i cPotentials [12].

F or i n t e g r a b l e s y s t e m s o f s e ve r a l de g r e e s o f f r e e dom t he s i t ua t i on i s a l mos t a ss a t i s f a c t o r y . T he l e a d i ng t e r m i n a n a s ym pt o t i c e xp r e s s i on is g i ve n by t heEins te i n-Br i l lo u in-Kel l e r (EBK) method us ing Mas lov ind ic es [13] . There appe ar s tobe no f unda m e n t a l d i f f i c u l t y i n t he w a y of e x t e nd i ng t he m e t hods u s e d f o r onede g r e e o f f r e e dom t o t h i s c a s e .

F or non i n t e g r a b l e s y s t e m s t he s i t ua t i on i s ba d .

I n p r a c t i c e t he EBK m e t hod a ppe a r s t o w or k a pp r ox i m a t e l y fo r t he r e gu l a r r e g i onand even for a pa r t o f the chaot ic r eg i on i f i t is in some sens e c lose to the r e gul a rr eg ion . This has been shown numer ica l ly for some model and molecular po te n t ia l s[14] . There has a l so been some e f f ec t ive use of c las s ica l and aemic las s ica l me th odsto es t ima te the r a t es o f p roc esse s invo lv in g h igh ly exc i ted a toms [15] .


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T h e r e i s e mp ir ic al e v i d e n c e a n d s o m e t h e o r y f o r t h e di f f e r en c e in s t r u c t u r e b e t w e e nt he e ne r gy s pe c t r a c o r r e s pon d i ng t o c om pl e t e l y r e gu l a r o r c om pl e t e l y c ha o t i csys t ems [16] and the be g inn ing s of a the ory of mixed sys t ems [17] . Much has beenw r i t t e n a bou t " q ua n t um c ha os " a nd t he r e ha s be e n m uch a r gum e n t a s to w he t he r i te x i s t s , bu t m uc h o f t h i s c on t r ove r s y s e e m s r a t he r ba r r e n .

Q u a n t a l p r o b l e m s ha ve a v e r y s u b t le c l a ss i c a l l im it . T h e n a t u r e o f t h i s l i m i t i sd e s c r i b e d b y s em ic la ss ic al m e c h a n i c s a n d t h e c o r r e s p o n d e n c e p r i n c i p l e , b o t h ofw h i c h a r e i n c o m p l e t e b e c a u s e t h e n a t u r e o f t he li mi t i s u n k n o w n f o r r ea li st icn o n i n t e g r a b l e s y s t e m s .

On e ca n legitimately ask the questions:

I. W h a t qu a n t a 1 p r o p e r t i e s b e c o m e c l a s s i c a l c h a o s f o r s m a l l h?

2 . W h a t c a n w e l e a r n a b ou t t h e p r o p e r t i e s o f s u c h q u a n t a l s y s t e m s f r o mt h e p r o p e r t i e s o f c l a s s i c a l c h a o t i c m o t i o n ?

A m a j o r d i f f i c u l t y i n a n s w e r i n g t h e s e q u e s t i o n s i s th a t w e d o n o t ye t f u l l yu n d e r s t a n d t h e p r o p e r t i e s o f c l a s s i c a l c h a o t i c m o t i o n .


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2: Hamil ton ian System of m Degrees of Freedom

DefinitionaWe cal l th ese mF Hamil t onian sys t ems.

The s ta t e o f such a sys tem i s r epr esen ted by a phase po in t X in a2m-dimens iona l phase space w i th gene ra l i sed coo rd in a te s and momenta :

X = ( q , p ) = (q z . . . . . qm; P: . . . . . p m ) .

The phase space need not be Eucl id ean: i t could be a mani fold.D e n o t i n g g r a d i e n t s b y :

a _ _ = < . . 8 _ . , a _ ~ ) 8 8 a9 " ap (~ I ' ". . . F~)8 q 8 q z 'a n d s c a l a r p r o d u c t s by ,

( 1 )

(2 )H a m i l t on ' s e qua t i ons f o r a r b i t r a r y m a r e :

= a H / a p ~ = - 8 H / a q .

T h e c h a n g e i n t h e v a l u e o f t h e H e ~ i l t o n i a n w i t h t i m e i s : -

(3 )

dH 8Hd -~ = 8 "t + < ~ , 8 H / S q > + = 8 H / S t + 0 ( 4 )

by (3) . I f the sys t em i s con se rv a t iv e , th a t i s au tonomo us , then H i s ind epen dent o ft ime and i t s va lue i s cons e rve d . I f th i s va lue i s an energy , then the ene rgyi s c ons e r ve d , bu t no t ne c e s s a r i l y o t he r w i s e . We us e " c o ns e r va t i v e " t o m e an t ha t t heva lue of the Hami l ton ian i s conse rved .

Area PreservingThe a r ea -pres e rv i ng prop er ty of Hami l ton ian sys t ems i s f und ament a l and i s no tr e s t r i c t e d t o 1F s y s t e m s . I t i s e qu i va l e n t to t he sym p l e o t i c p r ope r t y . F o r mFsys t ems w i th m>l i t is more impo r tan t than L iouvi l l e ts theorem on the pres e rva t ionof vo lume. The a r ea i s the bas ic Poincard invar i an t f rom which a ll o th e r s , inc lud ingthe vo lume, can be ob ta i ned . But i t s inva r ian ce does no t fo l low f rom the o the r s .


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A n a r e a i n t h e c l o s e d l o o p C o f t h e p h a s e s p a c e i s d e f i n e d f o r a g i v e n c o o r d i n a t esy st em (q,p) as the algebraic s um of the partial areas for qk, P k :

The figure illustrates this for 2F, wit h a four-dimensional Euclideanphase space.

. /"-C _ z..

/q~ Pl plane

r

J

Phase space :ql P~ q2P~Fig i. Comp one nts of areas for 2F system.

C i s a c l o s e d l o o p i n t h e p h a se s p a c e . S u p po s e e a c h p o i n t o f i t = o r e sa c c o r d i n g t o H a m i l t o n ' s e q u a t i o n s w i t h H a m i l t o n i a n H( q, p~ t ) , s o t h a tt h e w h o l e l o o p m o v es w i t h t h e H a m i l t o n i a n f l o w . T he n t h e c h an g e i nits area A is given by:

dA d d-~ = ~ c + ~

J C J Cf f= ~C - ~ < ap'aH dp>

: - { . ] c : oas H i s s i n g l e - v a l u e d . E q u a t i o n ( 6 ) i s f ro m i n t e g r a t i o n b y p a r t s an d( 7) f ro m H a m i l t o n ' s e q u a t i o n s .

( 6 )

( 7 )

( 8 )


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T h e a r e a i n a n y c l o s e d l o o p i s p r e s e r v e d i n a H a m i l t o n i a n f lo w.

F r o m e q u a t i o n (6) w e c a n d e r i v e a c o n v e r s e . If a r e a i s c o n s e r v e d f o r all t im e a n dfor all closed loops, (- p, q) is the gradie nt of som e single -valued function H(q,p,t)an d the sy st em is Hamiltonian.

Supp ose (q( t ) , p ( t ) ) r eprese n ts the s ta te o f a sys t em at t ime t , then thet r a n s f o r m a t i o n : -

T t : (q(o) ,p(o)) { > (q, (t), p(t) ) (9)

preserves area.

Canonical transformations

Let (q ,p) and (Q,P) be two coor d ina te sy s tem s or r ep rese n ta t ion s of po in t s X inpha s e s pa c e , o r t w o s t a t e s o f t he s y s t e m i n t he s a m e r e p r e s e n t a t i on .

Trans format ions :

( q , p ) ~ >

= < aF aH> 9F aH>aq' ~ + 8A 8B> (def) (12)[ A, a J : < ~ , ~ - < ~ p , ~F o r c a n o n i ca l t r a n s f o r m a t i o n s [ F , H ] i s i n d e p e n d e n t o f t h e r e p r e s en t a t i on , b e c a u s ei s i n d e p e n d e n t o f t h e r e p r es e n t a ti o n . S i n c e F a n d H a r e a r b i t r a r y t h i s a pp li e s toa n y [A, B ], a n d b e c a u s e o f t h i s i n d e p e n d e n c e t h e P o i s so n B r a c k e t ( PB ) isa n e w dyn ami cal variable. Somet imes, PB 's are trivial, for example, [A~ A].

P r o p e r t i e s o f P B ' s a r e :

P B I l i n e a r i t y i n A a n d B ]PB2 [A , B ] : - [ B , A ]PB3 [A,[B, C,]) + [B,[C,A]) + [C,[A,B]] = O

P B ' s d e f i n e a L i e a l g e b r a f o r d y n a m i c s l v a r i a b l e s.

Examples:

E l.

A is const ant for B is const ant fort h e B - f l o w t h e A - f l o w

(13)

( 1 4 )

E2. S y m m e t r y t h e o r y ( N o e t h e r )

F o r e x am p l e , c o n s i d e r a n g u l a r m o m e nt u m .

T h e a n g u l a r m o m e n t u m Li s a d y n a m i c a l v a r i a b l e w h i c h ,w h e n c o n s i d e r e d a s a H a m i l t o n i a n ,g e n e r a t e s r o t a t i o n a b o u t t h ez-axis.


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S u p p o s e H i s a H a m i l t o n i a n t h a t r e m a i n s u n c h a n g e d w h e n t h ep h a s e p o i n t s a r e r o t a t e d a b o u t t h e z - a xi s , ( n o t e th a t q k a n dPk must al_ll be rota ted). The n we have:

H i s u n c h a n g e d w i t h z - r o t a t i o n

H is c o n s t a n t f o r L z - f l o w(by g l )

Lz i s cons t an t fo r H- f low.the z-axis is conserved.

~Z

T h a t i s, a n g u l a r m o m e n t u m a b o u t

E 3 . F u n d a m e n t a l PB relations.

F o r a g i v e n c o o r d i n a t e s y s t e m , t h e y a r e g i v e n by :{qk, ql] = 0 = [P k, Pl} I

Jqk,Pl] = G k l ( i s )

T h e s e r e s u l t s a r e t r i v i a l a n d e a s y t o c h e c k i n ( q ,p )r e p r e s e n t a t i o n, b u t t h e y a r e i n d e p e n d e n t o f r e p r e s e n t a t i o n ,s o f o r a n y o t h e r ( Q, P) , w e c a n e r i t e : -

80 ' 8P0 = ( ~ 8pI~8 Q " 8 P -

~k l = < a q ~ 8PI>a q " 8 P

a _ S k a S _- ( 8 P ' a Q >_ < a _ P _ k ~ Q - ~ >8P '

( 1 6 )T h e s e a r e t h e f u n d a m e n t a l P B r e l a t i o n s f o r t h e r e p r e s e n t a t i o n s ( q, p)a n d (Q ,P ). k e y a r e o f t e n u s e d t o d e f i n e " c a n o n i c a l " i n s t e a d o f t h ei n t e g r a l a r e a - p r e s e r v i n g p r o p e r t y , b u t t h e y a r e n o t e x a c t l y e q u i v a l e nt .H o e e v e r , b e c a u s e t h e y a r e lo c a l a n d d i f f e r e n t i a l , t h e P B r e l a t i o n s e r em u c h e a s i e r t o c h e c k t h a n t h e a r e s - p r e s e r v i n g c o n d i t i o n a n d ca n b e u s e dt o o b t a i n c a n o n i c al t r a n s f o r m a t i o n s w i t h d e s i r e d p r o pe r t i e s .

- < S P ' 8 Q ~


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3 : I n v a r i a n t T o r i

Independent SystemsP a r t i c l e s c a n b e c o n f i n e d f o r v e r y l o n g t i m e s , w h e n t i m e i s m e a s u r e d i n u n i t s o f t h eb a s i c p e r i o d s . Q u e s t i o n s o f c o n f i n e m e n t a n d m a n y o t h e r q u e s t i o n s o f d y n a m i c s c a nb e d i v i d e d i n t o t w o p a r t s .

Q 1 ) A c c e s s i b i l i t y a n d i n v a r i a n t s e t s . G i v e n a n i n i t i a l s t a t e o f t h e s y s t e m , w h a to t h e r s t a t e s c a n i t r e a c h o r a p p r o a c h a r b i t r a r i l y c l o s e t o , g i v e n s u f f i c i e n t t i m e ?

Q 2 ) O r b i t s a n d m o t io n . W h e n d o e s i t g e t t o t h e s e st at ea p o r n e a r t h e m ?

C l e ar l y if t h e r e a r e c o n s e r v e d d y n a m i c a l v a ri a bl e s, t h e s y s t e m w i l l b e c o n f i n e d t or e g i o n s o f t h e p h a s e s p a c e f o r w h i c h t h e s e v a r i a b l e s h a v e t h ei r in it ia l v a l ue s .

C o n s i d e r a s i m p l e c a s e i n w h i c h t h e H a m i l t o n i a n i s a n e n e r g y a n d c a n b e e x p r e s s e da s a s u m o v e r 1 F H a m i l t o n i a n s o f t h e f o r m

mH ( X ) = H ( q , p ) - ~ H k ( q k , P k ) ( 1 7)

k = lT h e s i m p l e s t e x a m p l e i s m i n d e p e n d e n t c o n s e r v a t i v e s y s t e m s , b u t o t h e r s m a y b er e d u c i b l e t o t h i s f o r m .

T h e e n e r g y E o f t h e w h o l e s y s t e m i s c o n s e r v e d a n d t h i s c o n f i n e s t h e m o t i o n t o t h ee n e r g y s h e l l . T h e s e p a r a t e e n e r g i e s E k c a n a l s o b e c o n s e r v e d , b u t t h e s e a r e n o t s oc o n v e n i e n t t o u s e a s t h e a c t i o n v a r i a b l e s I k o f e a c h s y s t e m . E x c e p t f o r as e p a r a t r i x , t h e m o t i o n i s c o n f i n e d t o a c l o s e d c u r v e , o r t o p o l o g i c a l " c i r c l e " i n e a c hv a r i a b l e , w h e r e e a c h c i r c l e i s l a b e l l e d b y a n a c t i o n v a r i a b l e I k a n d p a r a m e t r i z e d b ya c o n j u g a t e a n g l e v a r i a b l e O k , p r o v i d e d t h e m o t i o n i s b o u n d e d .

T h e o v e r a l l b o u n d e d m o t i o n i s t h e r e b y c o n f i n e d t o m - d i m e n s i o n a l r e g i o n s i n p h a s es p a c e w h i c h a r e l a b e l l e d b y t h e a c t i o n v a r i a b l e I a n d p a r a m e t r i z e d b y t h e a n g l ev a r i a b l e e , w h e r e

I = ( I 1 , . . . . I m ) e = ( e l , . . . , e r a ) . ( 1 8 )o i s p e r i o d i c o f p e r i o d 2 ~ i n e a c h o k , a n d t h e r e g i o n i s t h e r e f o r e a n m - d i m e n s i o n a lt o r u s . T h e d i m e n s i o n a l i t i e s a r e v e r y i m p o r t a n t s o w e t a b u l a t e t h e m .


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Table o f Dimens ions

25

Number of de gr ee s 1 2 mof f r eedomP h a s e s p a c eE n e r g y s h e l lI n v a r i a n t t o r u sO r b i t

2 m2 m - 1

m

1

E a c h t o r u s is a n i nv a r i a n t s et . T o a n s w e r Q I w e o n l y n e e d t o k n o w t h e i n v a r i an tt or i, n o t t h e m o t i o n o n t h e m .

To answe r Q2 we need t he mot ion , which i s g i ven bye k ( t ) = W k t + e k ( o ) o r e ( t ) = tot + e ( o ) ( 1 9 )

w i t h p e ri o di c c o n di t i on s o n e. T h e a n g u l a r f r e q u e n c i e s w a r e g i v e n b yw = aH/ aI (20)

a s f o r t h e 1F c a s e. T h e m o t i o n i s a q u a s i p e r i o d i c f u n c t i o n o f t i me .I f t h e w k a r e n o t r a t i o n a l l y r e l a t ed , t h a t i s, t h e r e a r e n o i n t e g e r s s k s u c ht h a t

Ek Sk~tk = (s ,w ) = 0 (s ~ 0) (21)t h e n t h e p h a s e p o i n t p a s s e s a r b i t r a r i l y c l o s e t o e v e r y p o i n t o f t h e t o r u s : i te x p l o r e s t h e w h ol e t o r u s . F or i n t e g r a b l e s y s t e m s t h i s i s t h e t y p i c a l o r g e n e r i cca$ e .

O t h e r w i s e t h e r e i s a r e s o n a n c e : t h e m- t o r i a r e m a d e u p o f i n v a r i a nt t or i o f l o w e rd i m e n s i o n , t o w h i c h t h e m o t i o n i s c o n f in e d . T h e m - t o r i a r e d e g e n e r a t e .


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[ntegrable systems

An in t egra b le sys t em of m degr ees of f r eed om and Hami l ton ian H has m in t egra l s o fthe mot ion Fk , tha t a r e in involu t ion and ind epen den t , 8o for al l k , 1

[H, Fk] = 0 (F k an in te gr al ) (22a)[Fk, F1] = 0 (Fk, F 1 in in vo lu ti on ) (22b) = 0 ~ a = 0 (F k in de pe nd en t) . (23)

The mot ion of an in te grab le sys tem l ie s in a smooth mani fo ld of d imens ion m def i nedby

F = constant .I f t h i s m a ni f ol d i s c om pa c t a nd c onne c t e d , t he n i t i s a n i nva r i a n t t o r u s a nd t hemotion on i t i s quas ipe r iod i c w i th f r eque nc i es t~k (Liouvil le , Arnold: MathematicalMethods Chl0 , r e f . I ).

Th e involution condition (22b) implies that not ev er y s moo th m-t oru s in ph as e spac ec a n be a n i nva r i a n t t o r u s o f a n i n t e g r a b l e H am il t on i an s y s t e m . T he y a r e t he s pe c i a l' L a g r a ng i a n m a n i f o l d s ' w i t h t he p r ope r t y t ha t f o r a ny c l o s e d c u r ve Co in themani fo ld tha t can be de formed to a po in t w i thout l eav ing the mani fo ld , the r e la t ion

~c~ = 0 (on La gra ng ian ma ni fo ld ) (24)%, e

i s s a t i s f ied : The a r ea de f i ned on a Lagrangian mani fo ld i s ze ro .

( 3 9 )

w h e r e t h e i n t e g e r v e c t o r s h a s n o i n t e g e r f a c t o r e x c e p t + 1 a n d - 1 . F o r e x a m p l e( - 7 , 1 3) i s a p e r m i t t e d v a l u e o f s , b u t ( - 2 , 2 ) a n d ( 3, - 1 5 ) a r e n o t . T h e r e s o n a n c ei s g i v e n b y a s u m o v e r f u n c t i o n s o f e a c h r e s o n a n c e a n g l e , a n d c a n b e o b t a i n e d b y ar e s u m m i n g o f t h e F o u r i e r s e r i e s o v e r r a d i a l l i n e s f r o m t h e o r i g i n i n s - s p a c e . T h u sw e h a v e

r ( e ) : A o + [ - r E ( e s ) ( 4 0 )sw h e r e t h e s u m is o v e r p e r m i t te d s, e x c l u d i n g t h o s e w h i c h a r e o b t a i n e d f r o m e a c ho t h e r b y a s i g n c h a n g e , a n d

F s ( e s ) : ~. A j s e l j < s , e > ( 4 1 )J

w h e r e t h e j - s u m i s o v e r all i n te g er s .

T h e r e a r e a n i n f i n i t e n u m b e r o f r e s o n a n c e a n g l e s b u t f o r a n y o n e o f t h e e s t h e r ea r e r e p r e s e n t a t i o n s , a n d m a t r i c e s M , s u c h t h a t e s i s a n a n g l e v a r i a b l e .F o r m a n y d e g r e e s o f f r e e d o m , a s m a l l p e r t u r b a t i o n V ( e , I ) c a n b e a n a l y s e d i n t o as u m o v e r r e s o n a n c e t e r m s

V ( e , I ) = Vo + E s V ( e , I ) . ( 4 2 )I f t h e p e r t u r b a t i o n i s a f u n c t i o n o f o n l y o n e r e s o n a n c e a n g l e , t h e n t h e s y s t e m i si n t e g r a b l e , a n d h a s t h i s o n e r e s o n a n c e a t t h e i n v a r i a n t t o r u s f o r w h i c h

w s = < s , t r > = 0 . ( 4 3 )

T y p i c a l ly a ll t e r m s i n t h e e x p a n s i o n a r e n o n - z e r o , a n d t h e r e i s a n i s l a n d f or e v e r yd e g e n e r a t e t o r u s , a r b i t r a r i l y c l o s e t o e v e r y p o i n t i n p h a s e s p a c e .

I f I s i s a n a c t i o n f o r w h i c h t r - 0 , t h e n t h e a p p r o x i m a t e w i d t h o f t h e r e s o n a n c e i sg i v e n b y ( 3 6 ) , w h e r e i n ( 3 4 ) t h e v a r i a t i o n i s o v e r a l l t h e a n g l e v a r i a b l e s , a n d y i st h e c o m p o n e n t o f t h e n o n l i n e a r i t y i n t h e d i r e c t i o n o f t h e r e s o n a n c e a c t i o n .


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T h e a c tu al w i d t h s o f t h e r e s o n a n c e s a r e d e t e r m i n e d b y n o n l i n e a r in te ra ct io nsb e t w e e n t h e m , a n d m a y b e o b t a i n e d a p p r o x i m a t e ly b y e x p a n s i o n o r o t h e r it er at io nt e c h n i q u e s .

Reso nanc es a r e de f in ed in t e rm s of f r eq uenc ies t~ and pha se vo lumes in t e rms ofac t io ns I. The nonl in ea r i t y mat r ix Y i s the Jac obian mat rix of the t r an s form at i onbe t w e e n t he tw o s pa c e s a nd i s c r uc i a l f o r t he unde r s t a nd i ng o f pe r t u r ba t i ons o fnon l i ne a r s y s t e m s .

S om e i nva r i a n t t o r i a r e p r e s e r ve d unde r s u f f i c i e n t l y sm al l pe r t u r ba t i ons o f anon l i ne a r s y s t e m w h i c h ha s s u f f i c i e n t l y l a r ge non l i ne a r i t y . T hose i nva r i a n t t o r i a r ep r e s e r ve d w h i c h a r e s u f f i c i e n t l y f a r f r om a ll r e s ona nc e s w he n m e a s u r e d i n un i t s o ft he r e s ona nc e w i d th . S i nc e t he r e a r e a n i n f i n i t e num be r of r e s ona nc e s , w h i c ha p p r o a c h a rb it ra ri ly c l o se to a n y p o i n t i n p h a s e s p a c e r th is r e q u i r e s t h e r e s o n a n c ew i d t h s t o c o n v e r g e s u f f i ci e n t l y r a p i d ly w i t h t he o r d e r s o f t h e r e s o n a n c e . T h i srequi res V(e,I) to be sufficiently sm oo th for hig her Four ier (or reso nanc e) ter ms tob e s m a l l . K A M p r o v e t ha t f o r s uf fi ci en tl y s ma ll a n d s m o o t h p e r t u r b a t i o n s a po si ti veLiouville me as ur e of tori persists. This re quir es a stu dy of the interaction be tw ee nreso nance s, an d is crucial for question Q 1 o n invariant set s. Variational principlesc a n be u s e d t o s t udy i nva r i a n t s e t s a s d i s c us s e d i n M os e r ' s t a lk .

W h e n t w o r e s o n a n c e s i nt er ac t c ha ot ic m o t i o n i s u d u a l l y f o u n d . F o r w e a k i n t er ac ti onit is c o n f i n e d t o t h e n e i g h b o u r h o o d o f e a c h s e p a r a t ri x o f a n i nd iv id ua l r e s o n a n c e .F o r s t r o n g e r i nt er ac ti on it f il ls m o r e o f t h e p h a s e s pa ce . W h e n t w o o r m o r er e s o n a n c e s o v e r l a p t he c h a ot i c m o t i o n ( s to ch as ti ci ty ) t e n d s t o s p r e a d t h r o u g h o u tt h e r e g i o n o f p h a s e s p a c e o c c u p i e d b y all o f t h e m . T h i s w a s o b s e r v e d b y S y m o na n d S e s s l e r a n d h a s b e e n d e v e l o p e d i nt o a us e f u l t o o l f or t h e p ra ct ic al s t u d y o fe x t e n d e d c h a o s b y C h i r i k ov [ 4 ] . H o w e v e r , it m u s t b e u s e d w i t h c a r e a s t h e h i g h e ro r d e r p e r t u r b a t i o n s m a y b e n e c e s s a r y t o f in d s o m e o f t h e r e s o n a n c e w i d t h s a n dt h e r e a r e e v e n s pe ci al i n t e g ra b l e s y s t e m s f o r w h i c h t h e m e t h o d ( w r o n g l y , o f c o u r s e )predicts chaos'

C haos i s t he s ub j e c t o f m a ny c on t r i bu t i ons t o t h i s m e e t i ng , a nd i s e s s e n t i a l f o rques t ion Q2 on orb i t s , mot ion and l i f e t imes for conta inment . For cont a inm ent the rei s an es sen t ia l d i f f e r en ce be twe en 2F sys t ems and mF sys t ems w i th m~2. Look a tthe t ab le of d imens ions : f o r 2F the inv ar ia n t to r i o f 2D have one le s s d imens io nt ha n t he 3D e ne r gy s hel l , s o a s i ng le i nva r i a n t t o r u s c a n p r e ve n t pa r t i c l e s m ov i ngf rom one pa r t o f phas e space to ano ther , jus t a s a wal l (2D) can prev ent people


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from moving in 3D f rom the in s ide to the ou ts ide of a room. But for hi gh er m, thed i f f e r en ce in d imens ion i s 2 or more , so the to r i do no t fo rm an impa ssab l e ba r r ie r ,For 3F or mqre , the chaot ic r eg ion s a r e a l l jo ined upp (un les s t he re a r e spec ia lcons e rve d va r iab les} . I n th i s case the s low chaot ic mot ion for smal l pe r t urba t io ns ofin t egra b le sys t ems is ca l led Arnold d i f fu s ion and take s p lace a long the "Arnoldweb" of r eso nanc es in the pha se space . (See L ich ten berg an d L ieberman [1] .)

C o n f i n e m e n t

P r a c t i c a l c on f i ne m e n t i s de t e r m i ne d by m a ny f a c t o r s t ha t a r e no t t r e a t e d by pu r ec las s ica l Hami l tonian mechanics , inc l ud in g noise~ d i s s ipa t ion and q uant um e f f ec t s .Never the les sp the abs t r ac t p rob lem of con f in i ng the s ta t e o f a Hami l tonian sys t em toa g i ve n r e g i on R i n pha s e s pa c e ha s a s t r ong be a r i ng on t he p r a c t i c a l c on f i n e m e n tp r ob l e m, H am il t on i an c on f i ne m e n t t he o r y i s ne c e s s a r y bu t no t s u f f i c i e n t f o rp r a c t i c a l c on f i ne m e n t .

T he s t udy o f i n t e g r a b l e s y s t e m s a nd r e s ona nc e s f o r non i n t e g r a b l e s y s t e m s s ugge s t st he f o ll ow ing f a i r l y e v i de n t p r o c e du r e s f o r c on f i ne m e n t .

1. Make a ve ry de ta i l ed s tudy of the r eso nanc es ne a r the work in g r eg i on of phas espace.

2. Make pe r t urba t io ns as smal l and as smooth as poss ib le as func t io ns of angleva r i a b l e s .

3. In p ar t ic ula r con cen tra te on Vs(O s , I o) for r e s o n a n c e s sa) I n t e r na l t o w or k i ng r e g i on ( h i g he r o r de r )b) On b o u n d a r y of work ing r eg ion ( lower order ) .

Acknow ledgem ents , I should like to than k F ran ce V iva ld i f o r ve ry he lpfu ld i s c us s i o ns on c on f i ne m e n t p r ob l e ms .


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R E F E R E N C E S

1 . B o o k sV I A r n o l d a n d A A v ez " E r g o d i c p r o b l e m s o f c l a s s i c a l m e c h a n i c s " , W A B e n j a m i n ,N ew Y o r k ( 1 9 6 8 ) .J M o se r " S t a b l e a n d r an d om m o t i o n s i n d y n a m i c a l s y s t e m s , A n n a l s o f M a t h e m a t i c a lS t u d i e s N o 7 7" , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n ( 1 97 3) .V I A r n o l d " M e t h o d e s m a t h e m a t i q u e s d e l a m e c h a n i q u e c l a s s i q u e " , M i r , M o s c o w(1976); "Ma themat ical methods of class ical mechani cs", Springer, Ne w York (1978).I C P e r ci v a l a n d D R i c h a r d s " I n t r o d u c t i o n t o d y n a m i c s " , C a m b r i d g e U n i v e r s i t yPress (1982) (El ementa ry introduction ).A J L i e h t e n b e r g a n d M A L i e b e r m a n " R e g u l a r a n d s t o c h a s t i c m o t i o n . A p p l i e dM a t h e m a t i c a l S c i e n c e s 3 8 ", S p r i n g e r, N e w Y o r k ( 19 8 3) .J G u c k e n h e i m e r a n d P H o l m e s " N o n l i n e a r o s c i ll a t i o n s , d y n a m i c a l s y s t e m s, a n db i f u r c a t i o n s o f v e c t o r f ie ld s, A p p l i e d M a t h e m a t i c a l S c i e n c e s 42 ", S p r i n g e r, N e wYork (1983).R e v i e w sJ F o r d " S t o c h a s t i c b e h a v i o u r i n n o n l i n e a r o s c i l l a t o r s y s t e m s " i n L e c t u r e s i ns t a t i s t i c a l p h y s i c s e d W L S c h i ev e , S p r i n g e r , N e w Y o r k ( 19 72 ).J F o r d i n " F u n d a m e n t a l P r o b l e m s i n S t a t i s t i c a l M e c h a n i c s I I I " e d C o h e n , N o r t h -Holland, Ams ter dam (1975).K J Whi tem an Rep Prog Phys 40, 1033 (1977).M J Berr y in Am Inst Phys Con f Proc 46, 16 (1978).Y M Treve in Am Inst Phys Con f Proc 46, 147 (1978).G C a s a t i a n d J F o r d ( ed s ) " S t o c h a s t i c b e h a v i o u r i n c l a s s i c a l a n d q u a n t u mH a m i l t o n i a n s y s t e m s " , S p r i n g e r , B e r l i n ( 1 9 7 9 ) .D F E s e a n d e " L a r g e s c a l e s t o c h a s t i c i t y i n H a m i l t o n i a n s y s t e m s " , P h y s i c a S c r l p t aT 1 / 2 1 2 6 - 1 41 ( 1 9 8 2 ) .L G a r r i d o ( e d ) " D y n a m i c a l s y s t e m s & C h a o s : P r o c e e d S i t g e s 1 9 8 2 " , L e c t N o t e s i nP h y s 1 7 9, S p r i n g e r , B e r l i n ( 1 9 8 3 ) .

2 . E F e r m i , J P a s t a a n d S U la m " S t u d i e s o f n o n l i n e a r p r o b l e m s " L o s A lm o s R e p o r t L A-1 94 0 ( 1 9 5 5 ) ; F e r m i ' s C o l l e c t e d W o rk s No 2 6 6 , 9 7 8 - 9 8 8 ; " L e c t u r e s i n A p p l i e dM a t h e m a t i c s " o f t h e b Jd S 1 5 , 1 4 3 - 15 5 ( 1 9 7 4 ) .G C a s a t i a n d J F o r d ( s e e r e f 1 ) .G C a s a t i , J F o rd , F V i v a l d i a n d W M V i a s c h e r " O n e - d i m e n s i o n a l c l a s s i c a l , m a n y-b o d y s y s t e m h a v i n g n o r m a l t h e r m a l c o n d u c t i v i t y " s u b m i t t e d t o P h ys R ev L e t t s , ( 1 984 ) .

3 . A J D r a g t an d J M F i n n " I n s o l u b i l i t y o f t r a p p e d p a r t i c l e m o t i o n i n a m a g n e t i cd i p o l e f i e l d " J G e op h R e s 8 1 , 2 3 2 7 - 2 3 3 9 ( 1 9 7 6 ) .


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4 . K R S ym on a n d A M S e s s l e r " M e t h o ds o f r a d i o f r e q u e n c y a c c e l e r a t i o n i n f i x e df i e l d a c c e l e r a t o r s " P r o c CERN Sym p o n " h i g h e n e r g y a c c e l e r a t o r s a n d p i o n p h y s i c s "C ER N, G e n e v a 4 4 - 5 8 ( 1 9 5 6 ) .A Z i c h i c h i , K J o h n s e n a n d M H B l e w e t t " T h e o r e t i c a l a s p e c t s o f t h e b e h a v i o u r o fb e a ms i n a c c e l e r a t o r s a n d s t o r a g e r i n g s " C ER N, G e n e v a ( 1 9 7 7 ) .J M G r e e n e " M e t h o d f o r d e t e r m i n i n g a s t o c h a s t i c t r a n s i t i o n " J M a t h P h y s 2_00,1 1 8 3 - 1 2 0 1 ( 1 9 7 9 ) .B V C h i r i k o v " U n i v e r s a l i n s t a b i l i t y o f m a n y - d i m e n s i o n a l o s c i l l a t o r s y s t e m s "P h y s i c s R e p o r t s 5 2 , 2 6 3 - 3 7 9 ( 1 9 7 9 ) .M M o n t h a n d J C H e r r e r a ( a d s ) " N o n l i n e a r d y n a m i c s a n d t h e b ea m - b ea mi n t e r a c t i o n " A IP C o n f P r o c N o 5 7 , A m e r i c a n I n s t i t u t e o f P h y s i c s , New Y o rk ( 1 9 7 9 ) .

5 . M H e no n " E x p l o r a t i o n n u m e r i q u e d u p r o b l e m e r e s t r a i n t I V" B u l l A s t r S e r 3 , 1 ,4 9 - 6 6 ( 1 9 6 6 ) .

6 . J W isd om " T h e o r i g i n o f t h e K i rk w o o d g a p s : a m a p p i n g f o r a s t e r o i d a l m o t i o nn e a r t h e 3 / 7 a c o m m e n s u r a b i l i t y " A s t r o n J 8 7 , 5 7 7 -5 9 3 ( 1 9 8 2 ) .7 . S F D e r m o t t a n d C D M u r r a y " N a t u r e o f K i rk w o o d g a p s i n t h e a s t e r o i d b e l t "N a t u r e 30__!1, 2 0 1 - 2 0 5 ( 1 9 8 3 ) .8 . I C P e r c i v a l " S e m i c l a s s i c a l t h e o r y o f b o u n d s t a t e s " A dv C hem P h y s 3 6 , 1 - 6 1( 1 9 7 7 ) ; " H e g u l a r a n d i r r e g u l a r s p e c t r a o f m o l e c u l e s i n s t o c h a s t i c b e h a v i o u r i nc l a s s i c a l a n d q u a n t um s y s t e m s " ( s e e C a s a t i a n d F o r d , r e f 1 ) .

J F o r d , A d v C he m P h y s 2 4 , 1 5 5 ( 1 9 7 3 ) .M T a b o r , A d v C he m P h y s 4 6 , 7 3 - 1 6 2 ( 1 9 8 1 ) .S R i c e , A d v C he m P h y s 4 7 , 1 1 7 - 2 0 0 ( 1 9 8 1 ) .P B r u m e r , A d v C h e m P h y s 4_66, 2 0 1 - 2 3 8 ( 1 9 8 1 ) .J S H u t c h i n s o n , W P R e i n h a r d t , J T H y n es " N o n l i n e a r r e s o n a n c e s a n d v i b r a t i o n a le n e r g y f l o w i n m o d e l h y d r o c a r b o n c h a i n s " J C hem P h y s 7_99, 4 2 4 7 - 4 2 6 0 ( 1 9 8 3 ) .

9 . M V B e r r y a n d K M o u n t , R e p P r o g P h y s 3 _55, 3 1 5 ( 1 9 7 2 ) .1 0 . S e e 9 .1 1 . R B a l i a n a n d C B l o c h , A n n P h y s (N Y) 8 5 a , 5 1 4 ( 1 9 7 4 ) .1 2 . A V o r o s , A n n I n s t H e n r i P o i n c a r e A : P h y s i q u e T h e o r i q u e 3 99 , 2 1 1 - 2 3 8 ( 1 9 8 3 ) .1 3 . S e e P e r c i v a l , r e f 8 .1 4 . I C P e r c i v a l a n d N P o m p h r e y , M o l P h y s 3 1 , 9 7 ( 1 9 7 6 ) , J P h y s B 9 3 13 1( 1 9 7 6 ) M o l P h y s 3 5 , 6 4 9 ( 1 9 7 8 ) .

S M C o l w e 1 1 a n d I C P e r c i v a l , C he m P h y s 7_55 2 1 5 - 2 2 3 ( 1 9 8 3 ) .W P R e i n h a r d t " C h a o t i c d y n a m i c s , s e m i c l a s s i c a l q u a n t i z a t i o n , a n d m o d e-e n e r g y t r a n s f e r : t h e B o u l d e r v i e w " J P h y s Cham 8_66 2 1 5 8 - 2 1 6 5 ( 1 9 8 2 ) .


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1 5 . D A J o n e s , J G L e o p o l d , I C P e r c i v a l , J P h y s B : A t om M o l e c P h y s 1 3 ,3 1 - 4 0 ( 1 9 8 0 ) .B V C h i r i k o v , F M I z r a i l e v a n d D L S h e p e l y a n s k y " D y n a m i c a l s t o c h a s t i c i t y i nc l a s s i c a l a n d q u a n t u m m e c h a n i c s ", S o v i e t S c i e n t i f i c R e v i e w s C 2 2 0 9 - 2 6 7 ( 19 81 ).D L S h e p e l y a n s k y * ' S o m e s t a t i s t i c a l p r o p e r t i e s o f s i m p l e c l a s s i c a l l ys t o c h a s t i c q u a n t u m s y s t em s " . P h y s i c a 8 D 2 0 8 - 2 2 2 ( 1 9 83 ).

1 6 . M V B e r r y a n d M T a b or ~ J . P h y s i c s 100 3 7 1 - 3 7 9 ( 1 9 7 7 ) .M V B e r r y , 1 9 8 1 , A n n . P h y s . ( N . Y . ) 13__!1 1 6 3 - 2 1 6 ( 1 9 8 1 ) .M V B e r r y , S e m i c l a n s i c a l M e c h a n i c s o f R e g U l a r a n d I r r e g u l a rM o t i o n i n C h a o t i c B e h a v i o u r o f D e t e r m i n i s t i c S y s t e m s ( L e s H o u c h e s L e c t u r e sXXXVI, eds G. looss, R.H.G. Hell eman a nd H. Stora (North-Holland :Amsterdam) pp 171-271 (1983).M V B er r y , S t r u c t u r e s i n S a m i c ! a s s i c a l S p e ct r a : a Q u e s t i o n o f S c a l e i nThe Wave Parti cle Duali sm ( ed. S. Diner, D. Fargue, G. Lochak and F. Selleri)(D. Reidel: Dor dre cht) 231 -25 2 (1984).P Pech ukas, Phys. Rev. Lett. 51 943 -94 6 (1983).S.W. McDo nal d and A.N. Kaufman, Phys. Hey. Lett. 42 118 9-91 (1979)

17. M V Ber ry and M Robnik, J. Phys. A 17 2413- 2421 (1984).

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