b.a. khesin and yu.v.chekanov- invariants of the euler equations for ideal or barotropic...

8/3/2019 B.A. Khesin and Yu.V.Chekanov- Invariants of the Euler Equations for Ideal or Barotropic Hydrodynamics and Superc

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P h y s i c a D 4 0 ( 1 9 8 9 ) 1 1 9 - 1 3 1N o r t h - H o l l a n d , A m s t e r d am

I N V A R I A N T S O F T H E E U L E R E Q U A T I O N SF O R I D E A L O R B A R O T R O P I C H Y D R O D Y N A M I C S A N D S U P E R C O N D U C T I V I T YI N D D I M E N S I O N SB . A . K H E S I N a n d Y u . V . C H E K A N O VDifferential Equation Division, Departm ent of Mathematics, Moscow State University, Moscow 119899, U SSRR e c e i v e d 1 5 D e c e m b e r 1 9 8 8R e v i s e d m a n u s c r i p t r e c e i v e d 1 4 M a r c h 1 9 8 9A c c e p t e d 2 1 M a r c h 1 9 8 9C o m m u n i c a t e d b y V .I . A r n o i 'd

T h e H a m i l t o n i a n f o r m a l i s m f o r t h e Eu l e r e q u a t i o n s o f a n i d e a l f l u i d , s u p e r c o n d u c t i v i t y a n d a b a r o t r o p i c f l u i d o n aD - d i m e n s i o n a l R i e m a n n i a n m a n i f o l d i s p r o p o s e d . W e s h o w t h a t e a c h o f t h e s e e q u a t i o n s h a s a n i n f i n i te s er i es o f in t e g r a l s i f Di s e v e n ( " g e n e r a l i z e d e n s t r o p h i e s " ) a n d a t l e a s t o n e i n t e g r a l if D i s o d d ( " g e n e r a l i z e d h e l i c it y " ) . W e p r o v e t h a t t h e m a g n e t i ch y d r o d y n a m i c s i n t e g ra l f ( v , B ) # i s e q u a l t o t h e a v e r a g e l i n k i n g n u m b e r o f v e c t o r f ie ld s r o t v a n d B i n t e r m s o f t h e e r g o d i ct h e o r y . A l l t h e i n v a r i a n t s co n s i d e r e d a r e C a s i m i r d e m e n t s ( i.e . i n v a r i a n t s o f c o a d j o i n t a c t i o n ) o f t h e c o r r e s p o n d i n gi n f i n i t e - d i m e n s i o n a l L i e a l g e b ra s .

1. IntroductionTh e eq u a t i o n s o f an i n v isc id i n co mp ress ib l e f lu id

a r e H a m i l t o n i a n o n e s o n t h e o r b i ts o f th e g r o u p o fv o lu me-p re se rv in g d i f feo mo rp h i sms [ 1 , 2 , 4 ] . I ti s we l l k n o wn th a t fo r a two -d imen s io n a l f l o wth e re a re an i n f i n i t e n u mb er o f a rea i n t eg ra l s(ff(rotv)dZx) an d fo r a t h ree -d imen s io n a l o n eth e re i s a t o t a l h e l i c i ty i n teg ra l ( f ( r o t v , v )d 3 x ) .

Th e p r imary p u rp o se o f t h iS '~ l~p e r i s t o sh o wth a t t h e eq u a t i o n s o f t h e i d ea l f l u id h av e an i n f i -n i te n um be r o f in tegra ls on a n ! 'a rbi tra ry even-d i m e n s i o n a l m a n i f o l d M ( D = d i m M = 2r n ) a n d(a t l e a s t ) o n e i n t eg ra l fo r o d d D.

T h e s t a t e m e n t m e n t i o n e d w a s s e t t l e d f o r t h es t an d a rd R D wi th t h e h e lp o f so m e exp li c it co o r -d in a t e ca l cu l a t i o n s b y Se r re ( see re f . [1 5 ] ) . Hi sm e t h o d i s b a s e d o 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 ft h e E u l e r e q u a t i o n s p r o p o s e d b y O l v e r [ 13 ] . W esh o w th a t O lv e r ' s fo rmu la t i o n co in c id es wi th t h e

o n e g iv en b y A rn o r d ( see re fs . [2 , 4 , 7 ] ). Th u s w ean sw er t h e q u es t i o n ra i sed i n re f . [ 1 3 ] ab o u t t h ere l a t i o n o f t h e se two ap p ro ach es . No te t h a t o u rg en e ra l i za t i o n s d i f fe r g rea t l y f ro m th o se p ro p o sedb y Dez in i n re f . [ 6 ] (wh e re t h e o d d -d imen s io n a li n t eg ra l i s o b t a in ed ) .

Sec t i o n s 3 an d 4 co n t a in t h e mu l t i d imen s io n a lg en e ra l i za t i o n s o f su p e rco n d u c t i v i t y an d b a ro t ro -p i c f l u id eq u a t i o n s . We sh o w th a t t h e se eq u a t i o n sa r e H a m i l t o n i a n o n e s a n d f i n d t h e i r c o n -s e r v a t i o n l a w s a n a l o g o u s t o t h e h y d r o d y n a m i c a li n v a r i an t s . In t h e su p e rco n d u c t i v i t y ca se t h e p h asesp ace i s i d en t i f i ed wi th t h e d u a l sp ace (wh o seorig in i s d i sp laced) to the in f in i te -d imensional Liea lgebra of a l l d ivergence-free vec tor f ie lds .

Th e Eu le r fo rm o f t h e se eq u a t i o n s d i f fe r s f ro mt h e s t a n d a r d h y d r o d y n a m i c s e q u a t io n b y t h e C o r i-o i l s - t y p e t e rm. An a lo g o u s re su l t s fo r t h e t h ree -d i m e n s i o n a l s u p e r c o n d u c t i v i t y e q u a t i o n s w e r eo b t a i n e d b y H o l m a n d K u p e r s h m i d t ( s e e r ef . [1 1]) .

0 1 67-2 78 9 /8 9 /$0 3 .5 0 El sev i e r Sc i en ce Pu b l i sh e rs B .V.( N o r t h - H o l l a n d P h y s ic s P u b li s hi n g D i v i si o n )


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120 B.A. K hesin and Ytt V. Chekanov / lnvariants of the hydrodynamic equations

Th e co n f ig u ra t i o n sp ace o f t h e b a ro t ro p i c f l u idi s a semid i rec t p ro d u c t o f t h e d i f feo mo rp h i sms 'g ro u p an d t h e sp ace o f a l l fu n c t i o n s o n t h e man i -fo ld co n s id e red . Th e s imi l a r i t y o f t h e b a ro t ro p i cf l u id t o t h e i d ea l o n e can b e exp l a in ed b y i t s" i n c o m p r e s s i b i l i t y " i n t h e c o o r d i n a t e s c o n n e c t e dwi th t h e d en s i t y . So me re su l t s o n two -d imen s io n a lb a ro t ro p i c f l o ws can b e fo u n d i n an a r t i c l e b yHolm e t a l . [12] .

In t h e f i n a l s ec t i o n we d esc r ib e t h e e rg o d i cin t e rp re t a t i o n o f t h e t h ree -d imen s io n a l mag n e t i ch y d ro d y n a m ics i n teg ra l i n t h e sp i r it o f r e f. [ 5 ]. W es h o w t h a t t h e i n v a r ia n t f(B, v)d3x co in c id es wi thth e av e rag e l i n k in g n u mb er o f t h e ro to r v ec to rf ie ld ro t v an d the m agnet ic vec tor f ie ld B.On e o f t h e c ru c i a l p o in t s o f re f . [ 5 ] i s t h e e rg o d i cin t e rp re t a t i o n o f t h e t o t a l h e l i c i t y . Th e e rg o d i cin t e rp re t a t i o n o f mu l t i d imen s io n a l (mag n e to - ) h y -d ro d y n amics i n t eg ra l s i s an o p en an d i n t r i g u in gp r o b l e m .

Al l t h e i n t eg ra l s i n q u es t i o n a re i n v a r i an t s o ft h e co ad jo in t r ep re sen t a t i o n o f t h e co r re sp o n d in gLie groups ( the so-ca l led Casimir e lements) , i . e .t h e y d o n o t d e p e n d o n t h e p a r ti c u la r c h o ic e o f t h eHami l t o n i an . Th i s o p en s t h e way t o t h e i n v es t i g a -t i o n o f t h e n o n l i n ea r s t ab i li t y p ro b l em s b y Ro u th ' sm e t h o d [ 12 ]. W e a l s o th i n k t h a t i n f o r m a t i o n a b o u tth e o rb i t s can b e o f h e lp i n t h e s t u d y o f t h eC a u c h y p r o b l e m o f m u l ti d im e n s i o n a l h y d r o d y -n amics .No te t h a t t h e ex i s t en ce o f an i n f i n i t e n u mb er o fi n t eg ra l s fo r t h e f l ow o f an ev en -d imen s io n a l f l u idd o es n o t imp ly a co mp le t e i n t eg rab i l i t y o f t h ec o r r e s p o n d i n g h y d r o d y n a m i c s e q u a t i o n s . T h e i n -v a r i an t s co n s id e red o n ly d e f in e t h e co ad jo in t o r -b i t s (genera l ly speaking , in f in i te -d imensional ) , onwh ich t h e ev o lu t i o n t ak es p l ace . Fo r t h e eq u a t i o n so n t h i s man i fo ld -o rb i t t h e re i s a u n iq u e en e rg yin tegra l whi le the in tegrab i l i ty requi res an in f in i ten u m b e r o f t h e m .

I n f o r m a t i o n a b o u t t h e o r b i t s o f t h e c o a d j o i n tac t i o n (an d , h en ce , ab o u t t h e g eo me t ry ) o f t h eco r re sp o n d in g i n f i n i t e -d imen s io n a l L i e g ro u p s i sp r o b a b l y m o r e i m p o r t a n t i n i t s e l f t h a n i t s h y d r o -d y n a m i c s a p p l ic a t io n s .

2 . H y d r o d y n a m i c s o n t h e R i e m a n n i a n m a n i fo l dL e t M D d e n o t e a c o m p a c t R i e m a n n i a n m a n i -

f o l d ( w i t h o u t b o u n d a r y ) a n d g i t s v o l u m e f o r m(wh ich , i n g en e ra l , h a s n o co n n ec t i o n wi th t h efo rm in d u ced b y t h e m e t r i c s ( . , . ) ) . Th e eq u a t i o no f t h e i n co mp ress ib l e f l u id o n M is :

e = v p , (1 )d i v v = O ,

wh ere v an d p a re a t ime-d ep en d en t v ec to r f ie l dan d a fu n c t i o n o n M , (v , X7 )v d en o t e s t h e co v a r i -an t d e r i v a t i v e Vd o fo r R iem an n ian co n n ec t i o n .Theorem 1. Eq. (1) has( i ) the in tegra lI ( v ) = f Mu A ( d u ) " ( 2 a )

in t h e ca se o f an a rb i t ra ry o d d -d imen s io n a l man i -f o l d M ( D = 2 m + 1) ;

( i i ) an in f in i te number of in tegra ls

f f ( ( d u ) m ) #( v ) = ( 2 b )i n t h e ca se o f an a rb i t ra ry ev en -d imen s io n a l man i -f o l d M ( D = 2 m ) , w h e r e u i s a 1 -f o r m i n d u c e df r o m v b y t h e " l i f t i n g o f i n d i c e s " d e f i n e d b y t h eme t r i c s

v rxM,an d f i s an a rb i t ra ry fu n c t i o n o f o n e v a r i ab l e .Proof. Le t G b e t h e g ro u p o f a l l d i f feo mo rp h i smsp r e s e r v i n g v o l u m e # a n d ~ b e t h e L i e a l g e b r a o fa l l d ivergen ce-fre e vec tor f ie lds . Let ~2k and d~k_ 1d en o te t h e sp aces o f a l l k - fo rms an d exac t k - fo rmso n M, re sp ec t i v e ly .Lemma 1. (See a l so ref . [7] .) The re exis t s a na tu ra li so mo rp h i sm b e tween sp aces i f* an d 1 2 1 /d f~o. Th e


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B.A. Khesin and Yt t 1~. Chekan ov / lnvariants of the hyd rody namic equations 12 1

c o r r e s p o n d i n g p a i r i n g i s

=

Definition 1 . T h e s t a n d a r d l i n e a r P o i s s o n s t r u c t u r eo n i f * m a p s e a c h f u n c t io n a l F o n ~ * t o i tsH a m i l t o n i a n e q u a t i o n ( c a l l e d E u l e r - A r n o l ' d ' seq u a t io n , see r e f s . [ 1 , 4 ] ) ,

w h e r e v ~ i f , a n d th e f o r m u ~ I2~ i s an a r b i t r a r yr ep r e sen ta t iv e o f [u ] ~ I 2 1 /d J 2 0 .Proof . The e x a c t f o r m s c o r r e s p o n d t o t h e z e r of u n c t i o n a l , s i n c e

f M d f ( v ) p = f M d f A id x = O.T h e v a n i s h i n g o f t h i s i n t e g r a l f o l l o w s f r o m t h ef a c t t h a t t h e G - i n v a r ia n c e o f # i m p l ie s t h e d o s e -n e s s o f i v g a n d , h e n c e , th e e x a c t n e s s o f d f A i v# .T h u s , t h e d e f i n i t i o n o f t h e c h a n g e o f v a r i a b l e s i nt h e i n t e g r a l a n d t h e i n v ar i a n ce o f # i m p l y t h ec o i n c i d e n c e o f t h e c o a d j o i n t G - a c t i o n w i t h t h eG - a c t i o n o n t h e s p a c e o f 1 - fo r m s .Proposition 1 . T h e f o l l o w i n g i n t e g r a ls o n ~ * a r ei n v a r i a n t s o f t h e c o a d j o i n t a c ti o n :

( i ) i n ca se D = 2 m + 1

[ i l l = - - a d s F / s N [ u ] ,w h e r e t h e v a r i a t i o n a l d e r i v a t i v e 8F/8[u] E f~ isd e f i n e d b y t h e r e l a t i o n

+

f o r e a c h w ~ ~ * .N o t e t h a t o p e r a t o r a d * [ u] c o i n c id e s ( b y p r o p o -

s i t i o n 1 ) w i t h t h e o p e r a t o r Lv[u ] o f t h e L i e d e r iv a -t i v e d e t e r m i n e d b y t h e v e c t o r f i el d v o n M ( t h ed e f i n i t i o n i s c o r r e c t s i nc e L c o m m u t e s w i t h d ).Definition 2 . L e t ( . , . ) b e it h e R i e m a n n i a n m e t r i c so n M ( w h o s e v o l u m e f o r m i n g e n e r a l d if f e rs f r o mt h e g i v e n v o l u m e # ) . I t d e f i n e s a n o n d e g e n e r a t es c a l a r p r o d u c t o n & :

= ^ ( d u ) m ,

( i i ) i n ca se D - 2 m

an d , h en ce , an in v e r t ib l e o p e r a to r A : ~ -- -, f ~* ,c a l l e d t h e i n e r t i a o p e r a t o r ( s e e r e f . [ 4 ]) . I t m a p sth e v ec to r f i e ld v to c l a s s [u ] , sa t i s f y in g th e f o l lo w -i n g r e l a t i o n :

w h e r e f i s a n a r b i t r a r y f u n c t i o n o f o n e v a r i ab l e .Remark 1 . O b v i o u s l y , t h e g i v e n f u n c t i o n a l s a r ep r o p e r l y d e f i n e d o n ~ * , i .e . t h e y d o n o t d e p e n d o nt h e c h o i c e o f t h e r e p r e s e n t a t i v e i n c l a ss [ u ].

w h e r e u i s a n a r b i t r a r y r e p r e s e n t a t i v e o f [u ] . T h u s ,~ * i n h e ri t s f r o m ~ t h e n o n d e g e n e r a t e s c a la rp r o d u c t ( . , .)~ ,. W e d e f i n e o n ~ * t h e H a m i l t o n i a nf u n c t i o nz ( [ u l ) =

Proof. S i n c e t h e c o a d j o i n t a c t i o n c o i n c i d e s w i t ht h e c h a n g e o f v a r i a b l e s ( s e e l e m m a 1 ) , o u r s t a t e -m e n t f o l l o w s f r o m t h e c o o r d i n a t e - f r e e d e f i n i t i o no f t h e c o r r e s p o n d i n g i n t e g r a l s .

S i n c e H i s q u a d r a t i c , i t s v a r i a t i o n a l d e r i v a t i v e i s8 H / 8 [ u ] = A - t ( [ u ] ) . H e n ce , th e H a m i l t o n ia ne q u a t i o n f o r H c o i n c i d e s w i t h [ ~ ] = - L v[ u ] , w h e r e[ u ] = . , i v .


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122 B.A. K hesin and Y u. V. Chekanoo / Invariants o f the hydrod ynamic equationsI n t h i s e q u a t i o n w e m a y p a s s f r o m t h e c l as s e s t o

t h e p a r t i c u l a r r e p r e s e n ta t i v e s : ti = - L ~ u + d ~ .O p e r a t o r A - 1 t r a n s f o r m s i t i n t o t h e f o l l o w i n ge q u a t i o n o n i f :6 = - ( v , V ) v + d p ,

Rem ar k 3 . T h e i n t e g r a ls I ( [ u ]) a n d I f ( [ u ] ) d o n o tf o r m a c o m p l e t e s e t o f c o n t i n u o u s i n v a r i a n t s o fc o a d j o i n t o r b i t s . B y a n a l o g y w i t h t w o - a n d t h r e e -d i m e n s i o n a l c a s e s [ 5 ] , i t i s p o s s i b l e t o c o n s t r u c tp a r a m e t r i z e d f a m i l i e s o f o r b i t s w i t h e q u a l v a l u e so f t h e s e f u n c t i o n a ls .

c a l l e d t h e E u l e r e q u a t i o n o f t h e i d e a l i n c o m p r e s s -ib le f lu id ( see refs . [1 , 4 ] ) .

T h e e q u a t i o n 03 = - L v t 0 o n t h e s p a c e o f al le x a c t 2 - f o r m s ( t o - - - d u ) i s c a l l e d t h e H e l m h o l t zc u r l e q u a t i o n . T h e s a m e e q u a t i o n w i t h a m o r ec o o r d i n a t e - l i k e d e f i n i t i o n a p p e a r s a s O l v e r ' sH am i l t o n ia n f o r m u la t io n ( see eq . (3 .9 ) o f r e f. [1 3 ]) .

Remark 4 . T h e m a n i f o l d M m a y b e m u l t i -c o n n e c t ed . I n t h e n o n s i m p l y c o n n e c t e d c a s e t h ec o h o m o l o g i c a l c l a s s o f [ u ] i s a l s o a n i n v a r i a n t( c o m p a r e r e f. [ 3 ] ). O t h e r e x a m p l e s o f d i s c r e te i n -v a r i a n t s o f t h e E u l e r e q u a t i o n a r e t h e n u m b e r o fp o i n t s o n M w h e r e d u d e g e n e r at e s a n d t h e o r d e rso f i t s d e g e n e r a c y ( h e r e [ u ] = Av) .

L e m m a 2 . I n v a r i a n t s I ( [ u ] ) a n d I / ( [u]) f o r [u ] =A v a r e c o n s t a n t o n t h e t r a j e c t o r i e s o f t h e E u l e re q u a t i o n .Proof . T h e H a m i l t o n i a n c h a r a c t e r o f [ u ] i m p l i e si t s t a n g e n c y t o t h e o r b i t s o f t h e c o a d j o i n t G -a c t i o n o n f # * . H e n c e , Iy([u]) a n d I ( [ u ] ) a r e i t si n t e g r a l s . L e m m a 2 a n d , h e n c e , t h e o r e m 1 a r ep r o v e d .

F o r e x a m p l e , i n t h e s t a n d a r d m e t r i c s o f R 3,i n t e g r a l ( 2 a ) c o i n c i d e s w i t h

I ( v ) = f ( v , r o t v ) ~ t ,

Remark 5 . T h e m a n i f o ld M m a y b e n o n c o m p a c to r m a y h a v e a b o u n d a r y ( w e m a y c o n s i d e r M =R o ) . I n g e n e r a l, w e s h o u l d c o n s i d e r v e c t o r f i e ld st a n g e n t t o t h e b o u n d a r y .

T h e r e s t o f t h e s e c t i o n i s d e v o t e d t o t h e c a s e o ft h e o d d - d i m e n s i o n a l f l u i d . T h e o r e m 1 p r o v i d e s i nt h i s c a s e t h e e x i s t e n c e o f o n e i n v a r i a n t . T h e g e o -m e t r i c a l a p p r o a c h t o it s p r o o f a l lo w s t o o b t a i n t h ef o l l o w i n g s t a t e m e n t s .Corollary 1 . O n a n o d d - d i m e n s i o n a l m a n i f o l d ( D= 2 m + 1 ) th e cu r l v ec to r f i e ld (i .e . t h e k e r n e l o f( d u ) m ) i s " f r o z e n i n t h e f l u i d . "

a n d f o r R 2 i n t e g r a l ( 2b ) c o i n c i d e s w i t h

i : v ) = f I r o t v ) = f y a h o )w h e r e h a is t h e " f l o w f u n c t i o n " o f t h e v e c t o r f ie l dv r e l a t iv e to th e sy mp lec t i c f o r m/~ ( see r e f . [ 4 ] ) .Rem ar k 2 . T h e i n v a r i a n t ( 2 b) o f th e p l a n e - p a r a l le l2 m - d i m e n s i o n a l f lo w i n d u c e d b y ( 2 m - 1 )-d i m e n s i o n a l f l o w i s t r iv i a l, s i n c e ( d u ) m = 0 . T h e r e -f o r e , t h e r e d u c t i o n o f d i m e n s i o n g i v es n o i n t e g r a lsd i f f e r e n t f r o m ( 2 a ) .

Proof . Th e r o to r v ec to r f i e ld to i s d e f in ed b y th ec o n d i t i o n i o ,# = ( d u ) m .

Class [u ] ( an d , h en ce , ( d u ) m ) i s t r a n s p o r t e d b yt h e f l o w ; v o l u m e / t i s i n v a r i a n t a n d , h e n c e , t h ev ec to r f i e ld to i s a l so t r an sp o r t ed g eo me t r i ca l ly( i . e . i s " f r o zen in th e f lu id " ) .Corollary 2 . ( F o r D = 3 see r e f . [ 5 ] . ) Th e eq u a t io no f a n o d d - d i m e n s i o n a l i n c o m p r e s s i b l e f l u i d h a s ase t o f i n t eg r a l s :

I t ( v ) = ~ cu A ( d u ) ~


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B.A. K hesin and Y tt V. Chekanoo / lnoariants o f the hydrodynamic equations 12 3

w h e r e t h e i n t e g r a l i s t a k e n o v e r e a c h e r g o d i c c o m -p o n e n t o f t h e m o m e n t a r y c u r l ve c t o r f ie ld .

T h e p r o o f f o l lo w s i m m e d i a t e l y f r o m t h e S t o k e sf o r m u l a a n d f r o m t h e o b s e r v a ti o n t h a t t h e r e st ri c-t i o n o f ( d u ) " o n t h e b o u n d a r y o f a n y e r g o d i cc o m p o n e n t v a n is h es .Corollary 3 . L e t v d en o te a v ec to r f ie ld o f a f lo wo f a n i n c o m p r e s s i b l e f lu i d o n a n o d d - d i m e n s i o n a lm a n i f o l d a n d f a f u n c t i o n p r e s e r v i n g i t s v a l u e sf o r e ach f lu id p a r t i c l e , i .e . L f f = - 0 ( f o r exam p le , fi s o n e o f t h e L a g r a n g i a n c o o r d i n a t e s o f t h e f l u i d ) .T h e n t h e f l o w " i n t h e g e n e r a l p o s i t i o n " i s " c o m -p l e t e l y i n t e g r a b l e " i n t h e f o l l o w i n g s e n s e. T h e r ee x i s t D f u n c t i o n s in d e p e n d e n t o f M a l m o s t ev e ry -w h e r e , f o r m i n g a c o m p l e t e s et o f L a g r a n g i a n c o or -d i n a t e s . I n o t h e r w o r d s , k n o w i n g t h e s o l u t i o n o ft h e E u l e r e q u a t i o n ( t h e t i m e d e p e n d e n c e o f th ev e l o c it y ) a n d k n o w i n g o n e o f t h e L a g r a n g i a n c o o r-d i n a t e s , w e c a n c a l c u l a t e t h e o t h e r o n e s . S o , w ec a n d e f i n e t h e l o c a t i o n o f a n y f l u i d p a r t i c l e a t a na r b i t r a r y m o m e n t a v o i d i n g i n t e g r a t i n g t h e v e l o c -i ty .Proof . S in ce th e cu r l v ec to r f i e ld a~ an d f u n c t io n fa r e " f r o z e n i n t h e f l u i d " f o r t h e fl o w c o r r e s p o n d -i n g t o v , t h e d e r i v a t i v e o f f a l o n g ~ i s " f r o z e n "to o , i . e . i t i s t r an sp o r t ed a s a f u n c t io n . I t e r a t in gt h e p r o c e d u r e , w e o b t a i n f o r " g e n e r a l " f a n dt h e f u n c t i o n sf , Lo , , L ,oL , , . . . . . L o , . . . L , , , ,

2 mw h i c h a r e a l m o s t e v e ry w h e r e f u n c t io n a l l y i n d e-p e n d e n t a n d t h u s c a n b e u s e d a s L a g r a n g i a n c o o r-d i n a t e s .R e m a r k 6 . A t h r e e - d i m e n s i o n a l m a n i f o l d M i sd e c o m p o s e d i n t o a f i n i t e n u m b e r o f c e l l s b y t h ec r i t i c a l l e v el s o f s o m e f u n c t i o n f c o r r e s p o n d i n g t oa s t a t io n a r y f lo w v [1 , 5 ] . I n an y o f t h e se ce l l sd i f f e o m o r p h i c t o t h e p r o d u c t o f a t or u s a n d o f a ni n t e r v a l , f i e l d s v a n d ~ = r o t v a r e t a n g e n t i a l t ot h e l e v e l s u r f a c e s o f f ( w h i c h a r e d i f fe o m o r p h i c t oa t o r u s ) a n d d e f i n e t h e r e p e r i o d i c a n d q u a s i -

p e r i o d i c m o t i o n s . C h o o s i n g f a s o n e o f th eL a g r a n g i a n c o o r d i n a t e s , w e o b t a i n L j = 0 ( b yt h e t a n g e n c y c o n d i t i o n o f t h e v e c t o r f i e l d w a n dt h e l e v e l s u r fa c e s o f f ) , i.e . t h e s e c o n d L a g r a n g i a nc o o r d i n a t e c o n s t r u c t e d b y t h i s m e t h o d i s t r i v i a l .H e n c e , a n e r g o d ic b e h a v i o r o f th e f l ow o n s o m e o fth e to r i i s p o ss ib l e .R e m a r k 7 . I n th e ca se o f D = 3 th e in t eg r a l I ( v )h a s a n e r g o d i c i n t e r p r e t a t i o n a s t h e a v e r a g e s e l f -l i n k i n g n u m b e r o f r o t v ( s e e r ef . [ 5 ]) . F o r D > 3 as i m i la r i n t e r p r e t a ti o n i s u n k n o w n .

3 . T h e g ener a l i zed s uperconduct iv i ty equa t i o nT h e c o n n e c t i o n o f s u p e r c o n d u c t i v i t y t o t h e

e q u a t i o n s o f t h e i n c o m p r e s s i b l e f l u i d i s d u e t o t h ef a c t t h a t a t a h i g h d e n s i t y a n e l e c t r o n i c g a s i ss i m i l a r t o a f l u i d . I n d e e d , t h e r e p e l l i n g o f t h ep a r t i c l e s h a v i n g e q u a l c h a r g e s i n t h e e l e c t r o n i c a lc l u s t e r s m a k e s t h e g a s i n c o m p r e s s i b l e .

T h e e q u a t i o n o f ( n o n r e l a t i v i t i c ) s u p e r c o n d u c t i v -i ty in R 3 i s(~= - ( v , V ) v - v X B + V p , ( 3 )w h e r e v d e n o t e s a d i v e r g e n c e - f r e e f i el d o f t h ee l e c t r o n i c g a s v e l o c i t y , B a c o n s t a n t e x t e r n a l d i -v e r g e n c e - fr e e m a g n e t i c f ie l d, a n d s y m b o l t h ev e c t o r p r o d u c t f o r t h e s t a n d a r d m e t r i c s [ 9 ] .

W e d e f i n e t h e a n a l o g o f t h i s e q u a t i o n o n a na r b i t r a r y m a n i f o l d . L e t M b e a n a rb i t r a r y D -d i m e n s i o n a l m a n i f o l d w i th v o l um e # a n d t h eR i e m a n n i a n m e t r i c s g . W e s u p p o s e t h a t v i sd i v e r g e n c e - f r e e w i t h r es p e c t t o # , a n d B i s a" s t r i c t l y d i v e r g e n c e - f r e e " ( D - 2 ) - ve c t o r f ie l d w i t hr e s p e c t t o v o l u m e v r g , i .e . th e s u b s t i t u t i o n o f B i n

i s e x a c t: L n ~ / ~ = d a ( f o r e x a m p l e , i n t h e c a s eo f H 2 ( M ) = 0 , t h e c o n d i t i o n d i s k = 0 is suff i-d e n t ) . W e d e f i n e " t h e v e c t o r p r o d u c t " o f f ie ld va n d ( D - 2 ) - v e c t o r f i e l d B i n t h e s t a n d a r d w a y :v X B = , ( v ^ B ),


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124 B.A. Khesin and Yu. V. Chekanov Inoariants of the hydrodynamicequationsw h e r e * is t h e i s o m o r p h i s m o f k - a n d ( D - k ) -v e c t o r f i e ld s i n d u c e d b y t h e m e t r i c s [8 ]. W e c a l l (3 )t h e g e n e r a l i z e d s u p e r c o n d u c t i v i t y e q u a t i o n .T h e o r e m 2 . T h e m u l t i d i m e n s i o n a l s u p e r c o n d u c -t i v i t y e q u a t i o n ( 3 ) h a s

( i ) a n i n t e g r a l

I ( v ) = f M ( U + a ) A [ d ( u + a ) ] mi n t h e c a s e D = 2 m + 1 , a n d

( ii ) a n i n f i n i t e n u m b e r o f i n t e g ra l s

i n t h e c a s e D = 2 m , w h e r e a i s a 1 - f o r m s a t i s f y i n gt h e r e l a t i o n d a = i n x / ~ . T h e 1 - f o r m u i s o b t a i n e db y t h e " l i f t i n g o f i n d i c e s " o f f i e l d v w i t h t h e u s eo f m e t r i c s g .L e m m a 3 . T h e i n e r t i a o p e r a t o r A t r a n s f o r m s e q .( 3 ) i n t o t h e f o l l o w i n g e q u a t i o n o n ~ * :

( 4 ) o n th e q u o t i en t sp ace 1 2 t /d ~ 2 0 . F o r th e r ep r e -sen ta t iv e s o f t h e co n ju g a te c l a s se s eq . ( 4 ) r ead s :a = - L v ( u + a ) + d .L e m m a 3 i s p r o v e d .P roof o f theorem 2 . E q . (4 ) can b e wr i t t en a s

- L o [ u +s i n c e t h e c o n s t a n c y o f t h e m a g n e t i c f i e l d B i m -p l i es [ a ] ' = 0 . T h i s f o r m o f e q . (4 ) e n a b l e s u s t oc o n c l u d e t h a t [ u + a ] m o v e s a l o n g a n o r b i t o f t h ec o a d j o i n t G - a c t io n .

T h e f u n c t i o n a l s ( 2 a ) a n d ( 2 b ) e v a l u a t e d a t[ u + a ] a r e c o n s t a n t a l o n g th e o r b i ts o f th e s e p o i n t si n ~ * , s i n c e t h e G - a c t i o n i s a c h a n g e o f t h ev a r i a b l e s i n t h e e x p r e s s i o n [ u + a ] . T h e s e o r b i t sc o i n c i d e w i t h t h e a f f in e s h i f ts b y [ a ] o f t h e s t a n -d a r d o r b i t s o f t h e c o a d j o i n t a c t i o n o n ~ * . I np a r t i c u l a r , t h e s e f u n c t i o n a l s , e v a l u a t e d a t [ u + a ] ,w h e r e [ u ] = A v , a r e i n v a r i a n t a l o n g t h e s o l u t i o n o feq . (3 ) .

[ u ] ' = - L ~ [ u + a ] , (4 )w h e r e d a = i s ~fg , v = A - l [ u ] .P r o o f . T h e i n e r t i a o p e r a t o r A ( s ee s e c t i o n 1) is t h ec o m p o s i t i o n o f t h e " l o w e r i n g - o f -i n d i c e s " o p e r a t o ran d o f t h e p r o jec t io n ~ 1 ~ 1 21 /d 12 o- He n ce ,A ( v X B ) = A ( * ( v A B ) )

= == [ i o d a ] = I L i a ] . L ~ [ a ]

( t h e s e c o n d i d e n t i t y f o l l o w s d i r e c t l y f r o m t h e d e f -i n i t i o n o f * ( s e e r ef . [8 ] ) a n d t h e f i f th o n e f r o m t h eh o m o t h o p y f o rm u l a:

Corollary 4 . L e t [ u '] b e t h e n e w c o o r d i n a t e o n i f *d i f f e r e n t f r o m t h e s t a n d a r d o n e b y t h e s h i f t o f t h eo r ig i n t o t h e p o i n t [ - a ] : [ u '] = [ u + a ]. T h e n e q.( 4) i s a H a m i l t o n i a n o n e o n c o a d j o i n t o r b i t s w i t ht h e H a m i l t o n f u n c t i o n H = ([ u' - a ] , [ u ' - a ] )w i t h r e s p e c t t o t h e s t a n d a r d l i n e a r b r a c k e t o n an e w i f * .

I n d eed , eq . ( 4 ) i s [ u ' ] ' = L~[u'] f o r t h e n e w c o o r -d i n a t e [ u ' ] a n d A v = [ u ' - a ] .4. Th e m ult id im ension al barotropic f lu id

A b a r o t r o p i c f l ui d ( w h o s e p r e s s ur e d e p e n d s o n l yo n t h e d e n s i t y ) o n a m a n i f o l d M w i t h m e t ri c s g i sd e s c r i b e d b y t h e f o ll o w i n g s y s t e m o f e q u a t io n s o nt h e v e l o c i t y v a n d t h e d e n s i t y p :

L ~ - - i ~ d + d i ~ .T h e r e f o r e , i n a n a l o g y w i t h s e c t i o n 2 w e o b t a i n e q .

e = - ( v , v ) v + v h ( p ) ,+ d i v ( p v ) = 0 . ( 5 )


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B.A. Khesinand Y~ K Chekanov Invariantsof the hydrodynamicequations 125W e u s e t h e D - f o rm O o f d e n si t y O = p p~ D ( M ) , w h e r e p i s t h e d e n s i t y f u n c t i o n a n d # t h e

s t a n d a r d v o l u m e f o r m i n d u c e d b y t h e m e t r i c s / ~= v q .

T h e e q u a t i o n s o f t h e b a r o t r o p i c f l u i d a r e t h eH a m i l t o n e q u a t i o n o n t h e o r b i t s o f t h e c o a d j o i n tr e p r e s e n t a t i o n o f t h i s g r o u p , w h e r e t h e H a m i l t o nf u n c t i o n i s

Theorem 3 . E q . ( 5 ) o f t h e b a r o t r o p i c f l u id a d m i t s( i ) a n i n t e g r a l

I ( v ) = f M A ( d u ) "o n a n a r b i t r a r y o d d - d i m e n s i o n a l m a n i f o l d ( D =2 m + 1 ) ,

( ii ) a n i n f i n i t e n u m b e r o f i n t e g ra l s

o l o n a n a r b i t r a r y e v e n - d i m e n s i o n a l m a n i f o l d ( D =2 m ) , w h e r e 0 = p # a n d u is t h e 1 - fo r m o b t a i n e df r o m v b y t h e " l i f t i n g o f i n d i c es . "Proof . A h e u r i s t i c p r o o f o f t h e t h e o r e m i s b a s e do n t h e f a c t t h a t t h e f l u i d i s i n c o m p r e s s i b l e w i t hr e s p e c t t o t h e n e w v o l u m e 0 ( d e p e n d i n g o n t i m ea n d o n t h e i n i t i a l c o n d i ti o n s ) , s in c e t h e d e n s i t y i st r a n s p o r t e d b y t h e fl ow . T h u s , w e c a n a p p l y t h e o -r e m 1 , w h o s e a s s u m p t i o n s d e m a n d n o c o n n e c t i o nb e t w e e n m e t r i c s a n d v o l u m e . T h e r e s t o f t h e s e c -t i o n c o n t a i n s a m o r e d e t a i le d c o n s i d e r a ti o n o f t h eg e o m e t r y a n d a d i r e ct p r o o f o f o u r s t a t e m e n t .

T h e c o n f i g u r a t i o n s p a c e o f t h i s f l ui d i s a s e m i d i -r e c t p r o d u c t o f t h e d i ff e o m o r p h i s m s ' g r o u p a n d o ft h e s p a ce o f a ll f u n c ti o n s o n M : G = D i f f ( M )t,< C ~ ( M ) . R e c a l l t h a t t h e g r o u p s t r u c t u r e o n G i sd e f i n e d b y t h e f o r m u l a :(qo, a) * ( ~ , b ) = (p * ~b, qo,b + a )a n d t h e c o m m u t a t o r i n th e L ie a l g e b ra # i s[ ( v , a ) , ( t o , b ) l = ( [ v , t o ] , L , b - Loa) ,w h e r e qo, ~ ~ D i f f ( M ) , a , b ~ C ~ ( M ) , v , toV e c t ( M ) , a n d t h e s q u a re b r a c k e t s [ v, to ] d e n o t et h e o r d i n a r y c o m m u t a t o r o f t h e v e ct o r fi el ds .

p) --= fM[+p+ += h ( p ) .

T h e d u a l s p a c e t o V e c t ( M ) c a n b e i d e n t i f i e dw i t h t h e t e n s o r p r o d u c t o l ( n ) 12O(M) ( w i t ht h e n a t u r a l p a i r i n g : t h e v e c t o r f i el d i s s u b s t i t u t e di n t h e 1 - f o r m a n d t h e o b t a i n e d D - f o r m i s i n te -g r a t e d o v e r M ) . H e n c e , e le m e n t s o f # * a r e p a i rs( f l , 8 ) , w h e r e f l ~ ~ 2 X (M) ~ D ( M ) a n d 0 El ID(M) . T h e c o a d j o i n t a c t i o n o f t h e e l e m e n t(~p, a ) ~ D i f f ( M ) t~ C ~ ( M ) isAd ~ ;p , ,o ( f l , O) = (qo ,f l + d a qo ,O, qo ,O) .N o t e t h a t u = f l / O h a s t h e g e o m e t r i c s e n s e o f a1 - f o r m .Proposition 2 . T h e f u n c t i o n a l l ( f l , O) = fu ^( d u ) '~ i n t h e c a s e D = 2 m + 1 a n d t h e f u n c t i o n a l sI f ( # , 0) = f f (( d u ) " / O ) 0 i n t h e c a s e D = 2 m( wh er e u = f l/O ~ ~ 2 t ( M ) ) a r e i n v a r i a n t u n d e r t h ec o a d j o i n t G - a c t i o n o n # * .Proof.A d ~ , a , u = A d ~ , a ) ( f l ) = ~p,fl+daep,Oep,O

= ~ p , ( f l ) + d a = c p , u + d a ,i .e . , t h e 1 - f o r m i s t r a n s p o r t e d b y t h e f l o w m o d u l oa d i f f e r e n t i a l o f a f u n c t i o n ; t h e a c t i o n o n c l a s s[u] ~ I21 /d12 as we l l as on 0 ~ a2D i s r e d u c e d t ot h e c h a n g e o f v a r i a b l e s . N o w t h e p r o p o s i t i o n ( a sw e l l a s p r o p o s i t i o n 1 in s e c t i o n 2 ) f o l l o w s f r o m t h ec o o r d i n a t e - f r e e d e f i n i t i o n o f I a n d l f .

T h e i n e r t i a o p e r a t o r ,,1 : ~ ~ * , d e f i n e d b y t h em e t r i c s , a d m i t s t h e f o l l o w i n g e x p r e s s i o n : ( v , 0 )( u 0 , 0 ) , w h e r e 0 = 0/~ is t h e d e n s i t y f o r m a n d


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126 B.A. Khesin and Yu. V. Chekanov Invariants of the hydrodynamic equations1 - f o r m u i s o b t a i n e d f r o m v b y t h e " l i f t i n g o fi n d i c e s . " T h u s , t h e o r e m 3 i m m e d i a t e l y fo l lo w sf r o m p r o p o s i t i o n 2 in a n a l o g y w i t h s e c t io n 2.

5 . T h e e r g o d i c i n t e r p r e t a ti o n o f t h r e e - d i m e n s i o n a lm a g n e t o h y d r o d y n a m i c s

I n m a g n e t o h y d r o d y n a m i c s w e a s s u m e t h a t t h em a g n e t i c f i e ld B i s " f r o z e n " i n t h e i d e a l fl u i d o fi n f i n i t e c o n d u c t i v i t y f i l l i n g a m a n i f o l d M . T h ef l u id fl o w p r e s er v e s t h e v o l u m e / ~ o n M i n d u c e db y t h e m e t r i c s g . T h e v e l o c i t y f ie l d v a n d t h e" f r o z e n " m a g n e t i c f ie l d B ( d iv v = d i v B = 0 ) s a t-i s f y t h e s o - ca l le d m a g n e t i c h y d r o d y n a m i c a l e q u a -t i o n s :6 = - ( v , X T ) v + r o t B B + x T p,/ ~ = [ v, n ] ( 6)( t h e s e c o n d e q u a t i o n i s t h e d e f in i t i o n o f t h e" f r o z e n e s s " o f t h e f i e l d B , [ . , .] d e n o t e s t h e c o m -m u t a t o r o f t w o v e c t o r f i e l d s ) .

T h e s e e q u a t i o n s a r e c o n n e c t e d w i t h t h e L i ea l g e b r a ~ - = f f t~ i f * , w h i c h i s t h e s e m i d i r e c t p r o d -u c t o f t h e L i e a l g e b r a f f o f a l l d i v e r g e n c e - f r e ev e c t o r f ie l ds o n M w i t h th e v o l u m e / ~ a n d o f i tsd u a l sp ace ~ * . Th e d u a l sp ace ~-~-* co in c id es wi th~ * ~ f f a s a l i n e a r s p a c e .Proposition 3 . ( Co mp ar e wi th r e f s . [ 1 0 , 1 6 ] . )

( i ) T h e e q u a t i o n s o f t h e m a g n e t i c h y d r o d y n a m -i c s (6 ) a r e H a m i l t o n e q u a t i o n s o n t h e s p a c e ~ - *r e l a t i v e t o t h e s t a n d a r d P o i s s o n b r a c k e t .

( ii ) T h e q u a d r a t i c f o r m J ( [u ] , B ) = f~tu(B)l~o n ~ - * i s a n i n v a r i a n t o f t h e c o a d j o i n t r e p r e s e n t a -t i o n o f t h e c o r r e s p o n d i n g L i e g r o u p F = G ~ i f *(he re B ~ ~ , [u ] ~ I21 /d~2 = ~ * an d u ~ $2x is ar e p r e s e n t a t i v e o f [ u] ).Proof.( i) B y d e f i n i t i o n t h e c o m m u t a t o r i n f f t ~ ~ * i s[ ( , , ,

= ( [ v , =1 ~ , ad * [ /3 1 - a d * [ e l ) .

T h i s f o r m u l a i m p l i e s t h e f o l lo w i n g fo r m u l a f o r t h ec o a d j o i n t ~ a c t i o n o n ~ * :ad(* , t~ l ) ( [ u ] , B )

= ( L v [ u ] - L n [ a ] ,[ v , s l y ) . ( 7 )A n a r b i t r a r y m e t r i c s o n M ( s e e s e c t i o n 2 ) d e f i n e sa n i n e r t i a o p e r a t o r A : f f ~ & * , w h i c h d e t e r m i n e sa s c a l a r p r o d u c t o n ~ ' * . T h e c o r r e s p o n d i n gq u a d r a t i c f o r m i sH m a g ( [ u ], B ) -- ( [ u ] , A - l [ u ] ) + ( B, A B ) .T h e H a m i l t o n i a n e q u a ti o n s o n ~ * w i t h t h eH a m i l t o n f u n c t i o n H ~ a g i s , a c c o r d i n g t o ( 7 ) ,[ u ] ' = - L ~ [ u ] + L n [ b 1, ( 8 )w he re v = A - X[u], [b] = A B .

O n e c a n e a s i l y c h e c k t h a t i f t h e v o l u m e f o r m i sd e f in e d b y t h e m e t r ic s o n M , t h e n o p e r a t o r A - 1m a p s L n [ b ] t o r o t B B ( s ee s e c t i o n 3 ), a n d ,h e n c e , i t m a p s [ u ]" t o t h e v e c t o r f i e l d6 = - ( v , V ) v + r o t B B + V p .

T h i s e q u a t i o n t o g e t h e r w i t h t h e r e l a t io n s h =[ v , B ] ~ a n d d i v B = d i v v = 0 f o r m s t h e s y s t e m o fm a g n e t i c h y d r o d y n a m i c s e q u a ti o n s . T h u s , e q s. ( 8)a r e t h e i r i n v a r i a n t f o r m u l a t i o n .

( i i ) T h e i n v a r i a n c e o f t h e q u a d r a t i c f o r m J c a nb e v e r i f i e d b y t h e e x p l i c i t c a l c u l a t i o n u s i n g t h ef o l l o w i n g c o a d j o i n t a c t io n o f t h e g r o u p F :

* u B )d ( ~ , t . l ) ( [ ] ,=

N o t e t h a t t h e i n v a r i a n c e o f J w i t h r e s p e c t t o t h ea l g e b r a a c t i o n i m p l i e s o n l y t h e c o n s t a n c y o f J o nt h e c o n n e c t e d c o m p o n e n t s o f t h e c o a d j o i n t or b it s .T h i s e n d s t h e p r o o f o f p r o p o s it i o n 3 .Corollary 5. (See ref . [16] .) Th e va lue of fM(v, B ) I~i s p r e s e r v e d o n t h e t r a j e c t o r i e s o f t h e m a g n e t i ch y d r o d y n a m i c s e q u a t i o n s (8 ).


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B.A. Khesin and Yu. V. Chekanov lnvariants of the hydrodynamicequations 127I n d e e d , t h e s e t r a j e c t o r i e s a r e t h e i n v e r s e i m a g e s

o f t h e t r a j e c t o r ie s o f ( 8) o n t h e F - o r b i t s i n ~ - *( v = A - l [ u ] ) , o n w h i c h ( b y p r o p o s i t io n 3 ) f u ( B )i s p r e s e r v e d .

T h e m a i n g o a l o f t h i s s e c t i o n i s t o g i v e a ne r g o d i c i n t e r p r e t a t i o n o f t h e m a g n e t i c h y d r o d y -n a m i c s i n v a r i a n t s t h a t w e h a v e f o u n d . A t f i r s t w eg i v e t h e d e f i n i t i o n o f t h e a v e r a g e l i n k i n g n u m b e ro f t w o d i v e r g e n c e - f r e e v e c t o r f ie l d s [ 5] .

L e t M 3 b e a s i m p l y c o n n e c t e d m a n i f o l d w i thv o l u m e / t , a n d ~ a n d *1 t w o d i v e r g e n c e - f re e v e c t o rf i e ld s o n M ; g ~ a n d g ~ d e n o t e t h e i r p h a s e f lo w s ,F o r t w o p o i n t s x , y ~ M g i v e n , w e d e f i n e t h e" a s y m p t o t i c l i n k i n g n u m b e r " o f t h e t r a j e c t o r i e s o fg ~ a n d g ~ s t a r t i n g i n x a n d y , r e s p e c ti v e l y . F o rt h is p u r p o s e w e a t f i rs t c o n n e c t a n y t w o p o i n t s o nM b y a " s h o r t " p a t h A ( t h e c o n d i ti o n s i m p o s e do n t h e s h o r t p a t h w i l l b e d e s c r i b e d b e l o w a n d a r es a ti sf ie d a t " a l m o s t a n y " c h o ic e o f th e " s h o r t "p a t h s A ) .

W e s e l e c t t w o l a rg e n u m b e r s T a n d S a n d c l o s et h e s e g m e n t g ~ x ( 0 < t < T ) a n d g~y (0 < t < S )o f t h e tr a j e c t o r i e s i ss u in g f r o m x a n d y b y " s h o r t "p a t h s A ( g [ x , x ) a n d A ( g S y , y ) , s o t h a t w e o b t a i nt w o c l o s e d c u r v e s F = F r ( x ) an d F ' = F s ( y ). W ea s s u m e t h a t t h e s e c u r v e s a r e n o n i n t e r s e c t i n g ( t h i si s t r u e f o r a l m o s t a l l p a i r s o f x a n d y , a n d f o ra l m o s t a ll T a n d S ) . T h e n t h e l in k i n g n u m b e rN r , s ( X , y ) o f F a n d F ' is d e f i n e d .Defini t ion 3 . T h e a s y m p t o t i c l i n k i n g n u m b e r o ft h e p a i r o f t h e t r a je c t o r ie s g~x a n d g ~y i s d e f i n e da s t h e l i m i t

l im N r ' s ( X ' y)h ( x , y ) = r , s - . o ~ T S

Definit ion 4 . T h e a v e r a g e li n k i n g n u m b e r 2~ o f t w od i v e r g e n c e - f r e e v e c t o r f i e ld s i s

h = fM f~ ( x l , x2 ) t~ l# 2 N o w w e c a n s t a t e t h e m a i n r e s u l t , w h i c h i s am o d i f i c a t i o n o f t h e c o r r e s p o n d e n t s t a t e m e n t b yA r n o l ' d [ 5 ] :Theorem 4 . T h e a v e r a g e l in k i n g n u m b e r h o f t w od i v e r g e n c e - f r e e v e c t o r f ie l d s ~ a n d , / o n a s i m p l yc o n n e c t e d t h r e e - d i m e n s i o n a l m a n i f o l d M w i t hv o l u m e ~ c o i n c id e s w i t h

fMi~l~d - 1 ( i n / x ) .T h e c o n d i t i o n o f t h e v a n i s h i n g o f t h e d i v e r g e n c ef o r t h e v e c t o r f i e l d 7 ; o n M i s e q u i v a l e n t t o t h ec o n d i t i o n d i n ~ = 0 o r i n # = d r , w h i l e t h e i n t e g r a lf i t # A p e v i d e n t l y d o e s n o t d e p e n d o n t h e f r e e -d o m i n t h e c h o i c e o f i , ~ ~ 2 1(M ).Corol lary 6 . T h e m a g n e t i c h y d r o d y n a m i c s in v a r i-a n t f ( v , B ) # o n a s i m p l y c o n n e c t e d t h r e e - d i m e n -s i o n a l m a n i f o l d c o i n c id e s w i t h t h e a v e r a g e l in k i n gn u m b e r o f t h e v e c t o r f ie ld s r o t v a n d B . I n d e e d ,a p p l y i n g o u r t h e o r e m t o t h e v e c t o r f i el ds ~ = Ban d ~1 = ro t v ( an d u s i n g t h e r e l a t i o n [u ] = A v , d u= i ro tv /X) , we ob ta in

x A d - t ( i r o t v / . t ) = f i . z ^ d - l ( d u )

Q E D( T a n d S a r e t o v a r y s o t h a t F a n d F ' d o n o ti n t e r s e c t ) .

F u r t h e r w e a r e g o i n g t o p r o v e t h a t t h i s l i m i te x i s ts a l m o s t e v e r y w h e r e a n d i s i n d e p e n d e n t o ft h e s y s t e m o f " s h o r t " p a t h s A ( a s a n e l e m e n t o fL I ( M M ) ) .

N o t e t h a t , i n s p it e o f t h e d e p e n d e n c e o f v =A - l [ u ] o n t h e c h o i c e o f t h e m e t r ic s , t h e f ie l d r o t vi s d e f in e d u n a m b i g u o u s l y a s t h e k e r n e l f i e l d : o f -d[u].

T h e e r g o d i c i n t e r p r e t a t i o n o f f ( v , B ) t L a s o f a na v e r ag e li n k i n g n u m b e r o f r o t v a n d B i s s o m e -h o w u n e x p e c t e d , s in c e r o t v ( i n c o n t r a s t t o B ) is


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128 B.A. K hesin and Ym V. Chekanov / lnvariants of the hydrodynamic equations

n o t " f r o z e n " ( s ee e q . ( 8) ). T h e e v o l u t i o n c h a n g e st h e f i e l d [ u ] ( a n d , h e n c e , d [ u ] a s w e l l ) b y s o m ea d d i t i v e s u m m a n d , w h i c h d e p e n d s o n B , b u t i tt u r n s o u t t h a t t h e a v e ra g e li n k in g n u m b e r o f t h ek e r n e l f i e ld an d f i e ld B i s p r e se r v ed .Remark 8 . I n r e f . [ 5] t h i s r e su l t was p r o v ed f o r t h eco in c id e n t ~ an d ~1 ( f o r t h i s ca se th e in t eg r ab i l i t yo f t h e u n b o u n d e d f u n c t i o n s a r i s i n g i n t h e p r o o fw a s c h e c k e d ) . T h i s g i v e s a n e r g o d i c i n t e r p r e t a t i o nf o r t h e t o t a l h e l i c i t y i n t e g ra l .

T h e p r o o f p r o p o s e d b e l o w i s a m o r e i n v a r i a n tr e f o r m u l a t i o n o f t h e c o r r e s p o n d e n t a r g u m e n t o fr e f . [ 5 ] an d i t en ab le s to p r o v e th e in t eg r ab i l i t y int h e m o r e g e n e r a l c a s e .Proof. R e c a l l s o m e f a c t s a b o u t d o u b l e b u n d l e sa n d G a u s s - t y p e l i n k in g fo r m s .

Proof of proposition 4 . L e t D = d x + dy b e t h eo p e r a t o r o f t h e e x t e r n a l d e r i v a ti v e o n ~ 2 (M x M ) .Lemma 4.dx'-"~ = d o ~ .

I n d e e d ,[d x A ( x , y ) ] A p ( y ) = d x [A ( x , y ) A q 0 ( y ) la n d t h u s ,

f _~(x) [d~A(x , y ) ] A p ( y )= d ( L _,(xIA(X, Y) A eP(Y))"

Definition 5. K~ ~ 2 2 ( M X M ) i s c a l l e d a l i n k i n gf o r m o n a s i m p l y c o n n e c t e d m a n i f o l d M 3, i f f o ra n a r b i t r a r y p a i r o f n o n i n t e r s e c t i n g c l o s e d c u r v e sF 1 a n d F 2 w e h a v e :

fF~ K = N ( / ' x ' / ' 2 ) - F z c M M

w h e r e N(Fp/ '2) i s a l i n k i n g n u m b e r o f F x a n d F 2o n M , F l x F 2 = { ( x , y ) ~ M x M l x ~ I ' x , y ~F 2 } . T h e e x i s t e n c e o f s u c h a f o r m w i l l b e s e t t le dla t e r .Definition 6 . E a c h f o r m A(x, y) ~ O ( M x M) d e -t e r m i n e s a n o p e r a t o r A : f ~ ( M ) ~ ~ 2 (M ) s u c h t h a tcp(y) ~ f,~-kx)A(x, y) A cp(y) , wh ere ~r: M X M

M i s t h e p r o j e c t i o n o n t h e fi r st c o m p o n e n t a n dt h e i n t e g r a t i o n i s p e r f o r m e d o v e r t h e f i b r e s o f t h i sp r o j e c t i o n .Proposition 4 . A n o p e r a t o r g c o r r e s p o n d i n g t ot h e l i n k i n g f o r m i s a G r e e n o p e r a t o r , i . e . , a no p e r a t o r i n v e r s e t o t h e e x t e r n a l d e r i v a ti v e : i f t k =d cp an d p ~ Q X (M) , t h en p = g ( tk ) + d f ( th es u m m a n d d f p o i n t s t h a t q~ c a n b e d e fi n e d b yo n l y m o d u l o a f u l l d i f fe r e n ti a l ).

Lemma 5 . I f A i s a 1 - f o r m i n t h e v a r i a b l e y , t h e nd y A = A o d .I n d e e d ,

f,,-~(x) [ d yA(X , y ) ] A r p ( y )= L _ I(xA(X, y) A dcp(Y)

Lemma 6 . T h e e x t e r n a l d e r i v a t i v e o f a l i n k i n gf o r m i s D K = 3 + f l , w h e r e 8 i s th e 8 - f o r m o n t h ed i a g o n a l i n M x M a n d f l ~ Q 3 ( M x M ) i s a li n -e a r c o m b i n a t i o n o f d o s e d ( a n d , t hu s , e xa c t) f o rm sw i t h r e s p e c t t o e a c h o f t h e t w o a r g u m e n t s .Proof.N(F 1, F2) = fqxr2 K= fa-~(_r,lxr2)D K

= f ~ D K .a-tFO/"2O n t h e o t h e r h a n d , b y t h e d e f i n i t i o n o f t h e l i n k i n gn u m b e r a s t h e i n t er s e c ti n g n u m b e r o f t h e c y c l e F 2w i t h a f i lm 0 -1 F 1 ( w h o s e b o u n d a r y is F t ) w e


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B.A. Khesin and Ym V. Chekanov lnvariants of the hydrodynamic equations 1 29o b t a i n

N ( r , , / ' 2 ) = [ 6 ., ( a - ' r , )x r2T h e n t h e s t a t e m e n t f o ll o w s f r o m t h e f a c t th a t

f( /3=0.o - ' r , )r2Remark 9 . In sp i t e o f t h e n o n v an i sh in g o f t h ec o h o m o l o g i c a l c l a s s o f 6 ( s i n ce t h e d i ag o n a li n M M i s n o t a b o u n d a r y ) w e c a n c h o o s e f lsa t i s fy in g t h e v an i sh in g co n d i t i o n fo r t h e co h o mo -lo g i ca l c la s s o f 6 + f l ( t h e d i ag o n a l can b e d eco m -p o s e d i n t h e s u m o f h o m o l o g y g r o u p s ' g e n er a to r s ).Th u s , we h av e p ro v ed t h e ex i s t en ce o f a l i n k in gfo rm K a s t h e so lu t i o n o f t h e eq u a t i o n :D K = 0 ~ H 3 ( M M ) .

T o f in is h t h e p r o o f o f p r o p o s i ti o n 4 w e p a s s int h e e q u a t i o n D K = 6 + f l f r o m t h e f o r m s t o t h eo pe ra to rs : D K = 6 + f l o r d x K + d y K = 6 + f l .S in ce t h e 6 - fo rm co r re sp o n d s t o t h e i d en t i t y o p e r -a t o r , a n d t h e i m a g e o f t h e o p e r a t o r / ~ b e l o n g s t ot h e s p a c e o f t h e e x a c t f o r m s b y l e m m a s 4 - 6 w eo b t a i nd o / ~ + / ~ o d- -- i d + d * ~ .A p p ly in g t h e se o p e ra to rs t o t h e fo rm cp we g e td o /~ (~ 0 ) + /~ (d (p ) - - q~ + d o -~(p ).S ince ff = d~p , the final f o r m u l a i sp = / ~ ( t } ) + d f . Q E DL e m m a 7. There i s the l ink ing form K ( x , y ) h av -i n g a p o l e o f d e g r e e 2 o n t h e d i a g o n a l o f M M ,i .e . i t s coeff ic ien ts g row as c / I x - y l 2 (we u se ana r b i t r a r y m e t r i c s o n M M i n d u c e d b y a m e t r i c son M).

Proof. I n f a c t , th e l in k i n g n u m b e r o f F 1 a n d F2co in c id es wi th t h e l i n k in g n u m b er o f F 1 F 2 an do f t h e d i ag o n a l i n M M. We id en t i fy a n e ig h -b o u rh o o d o f t h e d i ag o n a l i n M M an d a n e ig h -b o u r h o o d o f z e r o - s e c t i o n i n n o r m a l b u n d l e o f t h ediagonal (i .e . in T i M ) b y t h e e x p o n e n t i a l m a p .Th en i n ev e ry f i b re ( t h a t i s a n e ig h b o u r h o o d o f0 ~ R 3 ) w e c o n s i d e r t h e s t a n d a r d l in k i ng f o r mwi th t h e p o in t 0 . Th i s l i n k in g fo rm co in c id es wi ths u b s t i t u t i o n o f v e c t o r f i e l d X T ( - 1 / r ) = ( V r ) / r 3(wi th a p o l e i n 0 o f d eg ree 2 ) t o t h e s t an d a rdv o lu me fo rm in a f i b re . At l e a s t , l e t t h i s fo rm b ev an i sh ed o n v ec to rs p a ra l le l t o M c T _ M. So , t h ec o r r e s p o n d i n g l i n k i n g f o r m i n M M h a s a p o l eo f d eg ree 2 o n t h e d i ag o n a l. Q EDCorollary 7. The l ink ing form K is in tegrab le , i . e .t h e v a l u e o f K e v a l u a te d a t a n y t w o s m o o t h v e c t o rf i e ld s i s an i n t eg rab l e fu n c t i o n o n M M (KL I ( M M ) ) .

In d eed , t h e co d imen s io n o f t h e d i ag o n a l i n M M e q u a l s 3 a n d t h e o r d e r o f g r o w t h o f K n e a rth e d i ag o n a l eq u a l s 2 .

No te t h a t a l l t h e p rev io u s a rg u men t s o n l i n k in gfo rms h o ld (wi th so me ev id en t mo d i f i ca t i o n s ) fo rt h e m a n i f o l d s o f a n y d i m e n s i o n . T h e f u r t h e r c o n -s id e ra t i o n s a re e s sen t ia l l y th ree -d imen s io n a l. Th u s ,we h av e two d iv e rg en ce - f ree f i e l d s ~ an d 7 /o n M,e q u i p p e d w i t h t h e v o l u m e f o r m # .Definition 7. (See ref . [5 ] .) A system o f short pa thsjo in in g t h e p o in t s x an d y ~ M i s a sy s t em o fp a t h s d e p e n d i n g i n a m e a s u r a b l e w a y o n x a n d y ,su ch t h a t t h e i n t eg ra l s o f t h e l i n k in g fo rm K fo rev e ry p a i r o f n o n in t e r sec t i n g p a th s o f t h e sy s t em,an d a l so fo r an y n o n in t e r sec t i n g p a i r (p a th o f t h es y s t e m , u n i t a r y t i m e s e g m e n t o f p h a s e c u r v e o ff i e ld s ~ o r ~1) a re b o u n d ed i n d ep en d e n t ly o f t h ep a t h s b y a c o n s t a n t C .

Su ch sy s t ems ex i s t s i n ce t h e i n t eg ra l o f K fo rt h e p a i r o f p a t h s r e m a i n s b o u n d e d w h e n t h e s ep a t h s a p p r o a c h " t r a n s v e r s a l l y . "

L e t g~x and g~y be the t ra jec tor ies o f the f ie ldsan d 7/ s t a r ti n g f ro m x a n d y fo r th e t ime


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130 B . A . K h e s in a n d Y u . V . C h e k a n o v l n v a r i a n t so f t h e h y d r o d y n a m i c q u a t io n si n t e rv a l s 0 < t < T a n d 0 ~ t < S , A x a n d A y b et h e c o r r e s p o n d i n g s h o r t p a t h s d o s i n g t h e s e t r aj e c -t o r i e s . N o w

X ( x , y ) = 1 Kl i m T S f ( g ~ x + A x ) ( g ~ y + A yT , S. . .. - , o1 KT , l i m ~ ~ f ~ x g ; y "

T h i s e q u a l i t y o f l i m i ts f o l lo w s f r o m t h e b o u n d n e s so f t h e i n t e g r a l o v e r t h e s h o r t p a t h s . T h e n

f f M X M h ( X ' Y ) t i x l ~ y= l i r ax Y T , S - .* o o= f ~ S ~ " r,s-+ = lim~ - - ~ S o f o ( i , i ~ l K ) d t d s

w h e r e f o a n d S d e n o t e t h e i n t e g ra l s o f t h e f u n c -t i o n i ~ i , i K a l o n g t h e t r a j e c t o r i e s g ~ x a n d g ~ y . B yt h e B i r k h o f f e r g o d ic t h e o r e m w e c a n p a s s f r o m t h et i m e a v e r a g e s t o t h e s p a c e a v e r a g e :

f f M M ~ ( X , Y ) l i x l ~ Y = f M ~ x f M ~ Y ( i , i , K )( r e c a l l t h a t K ~ L I ( M M ) ) . R e c a l l i n g t h e d e f t -n i t i o n o f th e a v e r a g e li n k i n g n u m b e r h a s t h ea v e r a g e v a l u e o f h ( x , y ) o v e r M M a n d s h i f t i n gt h e o p e r a t o r s o f s u b s t i t u t i o n i~ a n d i n t o t h ef o r m s g x a n d / . t y w e o b t a i n t h e f o l l o w i n g :

~ = f f M x M h ( X , y ) l X x l i Y = f M ~ x f M ~ Y ( i , i , K )

= f u ~ A g ( i ~ / ~ ) .

B y p r o p o s i t i o n 4 I ~ ( i , ll i ) = d - a ( i n # ~ ) m o d u l o a ne x a c t f o r m , i . e . X = f i~ /~ A d - l ( i n / x ) . T h e p r o o f o ft h e o r e m 4 is c o m p l e te d .

I t w o u l d b e v e r y i n t e r e s t i n g t o f in d t h e m u l t i d i -m e n s i o n a l g e n e r a l i z a t io n o f th i s e r g o d i c i n t e r p r e -t a ti o n . P r o b a b l y , t h is i n t e r p r e t a t i o n i s c o n n e c t e dw i t h t h e s y m p l e c t i c p r o p e r t i e s o f t h e s p a c e o f t h et r a j e c t o r i e s o f a d i v e r g e n c e - f r e e v e c t o r f ie l d a n do u r i n t e g r a l i n v a r i a n t s w o u l d b e i n s o m e s e n s e t h e" a s y m p t o t i c " M a s l o v i n di ce s .

AcknowledgementsW e a r e p r o f o u n d l y g r a te f u l to V . I. A r n o l ' d f o r

t h e p r o p o s a l o f t h e p r o b l e m s a n d f o r m a n y f r u i t -f u l d i s c u s s i o n s o f t h e id e a s t r e a t e d h e r e . T h e r e -s u l ts o f s e c t i o n 2 ar e o b t a i n e d j o i n t l y w i t h V . Y u .O v s i e n k o [ 14 ]. W e a r e g r a t e fu l t o h i m a n d a l s o toI .S . Z a h a r e v i c h a n d V . V . F o c k f o r h e l p f u l d is c u s -s i o n s .

References[1] V .I. Ar nol'd , Su r la g6om& rie diff~rentielle des groupes d eLie de dimension infinite et ses applications ~ rhydrody-namique des fluides parfaits, Ann. Inst. Fourier 16 (1966)319.[2] V.I. Arnol'd, Hamilton character of the Euler equations ofsolid body and ideal fluid, Usp. Mat. Nauk 24 (3) (1969)225 (in Russian).[3] V.I. Arn ol'd, O n one-dimensional cohomology of Lie alge-bra of divergence-free vector fields and on rotation num-ber of dynamical systems, Funct. Anal. Appl. 3 (1969) 4.[4] V.I. Arnol'd, Mathematical Methods of Classical Mechan-ics (Springer, Berlin, 1978).[5] V.I. Arnol'd, The asymptotic Hopf invariant and its appli-cations (1974) (in R ussian), English transl. Sel. Math. Soy.5 (1986) 327.[6] A.A. Dezin, Invariant forms and some structure proper-ties of the Euler equations of hydrodynamics, Z. Anal.Anwend. 2 (1983) 401 (in Russian).[7] J. Marsden and A. Weinsteia, Coadjoint orbits, vortices,and Clebsch variables for incompressible fluids, Physica D7 (1983) 305.[8] B.A . Dub rovin, S.P. Novikov and A.T. Fomenko, Moderngeom etry (Nauka, Moscow, 1979) (in Russian).[9] R.P. Fey nm an, Statistical Mechanics (Benjamin, NewYork, 1972).[10] D.D. Holm and B.A. Kupershmidt, Poisson brackets andClebsch representations for magnetohydrodynamics, m ul-tifluid plasmas, and elasticity, Physica D 6 (1983) 347.[11] D.D. Holm and B.A. Kupershmidt, Poisson structures ofsuperconductors, Phys. Lett. A 93 (1983) 177.


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B.A. Khesin and Ym 1I. Chekanov / Invariants of the hydrodynamic equations 13 1[ 1 2 ] D . D . H o lm , J . E . M arsd en , T . R a t iu an d A . W ien s t e in ,N o n l in ea r s t ab i l i t y co n d i t i o n s an d a p r io r i e s t im a tes fo rbaro trop ic hydrodynamics , Phys. Let t . A 98 (1983) 15 .[13] P . J. O lver , A non l inea r Ham il ton ian s t ructu re fo r theEuler equat ions , J . Math . Anal . App l . 89 (1982) 233 .[14] V.Yu . Ovsienko , B .A. Khesin and Yu .V. Chekanov , In te-

g ra l s o f t h e E u le r eq u a t io n s o f m u l t id im en s io n a l h y d ro d y -n am ics an d su p e rco n d u c tiv i ty , Z ap i sk i L OM I 1 72 (19 8 9)

105 (in Russian), Engl. transl, in J . Sov. Math. , to appear.[15] D. Serre , Invar ian ts e t d~g~n6rescence symplect ique del '~quat ion d 'Eu ler des f lu ides parfai ts incompress ib les ,C. R. Acad. Sci. (Paris) Set. A. 298 (1984) 349.[16] S .V. Vish ik and F .V. Dolzansk i i , Analogs o f the Eu ler-P o i s so n eq u a t io n s an d m ag n e to h y d ro d y n am ics eq u a t io n sco n n ec ted w i th L ie g rou p s, D o k l . A k ad . N au k U S S R 2 3 8(1978) 1032.

b.a. khesin and yu.v.chekanov- invariants of the euler equations for ideal or barotropic...

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