3 9 1 3 % %a wTMk%*% Vol. 13, No. 4

1990 & ~ O R ACTA MATHEMATICAE APPLICATAE SINICA OS~. , 1990

Hamilton g3*

SOME HAMILTONIAN OPERATORS IN INFINITE DIMENSIONAL HAMILTONIAN SYSTEMS

(Comparing Centre of Academia Sinice)

Abstract

In this paper some Hamiltonian operators with special forms are presented. Firstly, the sun operators of a class of u-linear differential operators of the first order and constant coef- ficient differential operators are discussed, and the necessary and sufficient conditions for then? to be Hamiltonian are obtained. Based on these conditions, one typical case is analysed. Next,. three Hamiltonian operators are generated directly from infinite dimensional Lie algebras. Thirdly, a kind of matrix differential operators corresponding to the structural' functions '01 the second degree is constructed and an algebraic description for them to be Hamiltonian is proposed.

Z€@PB%%EPE% Hamilton %%ElIFA% @ ," 2$ Hamilton $ $ ~ ~ ( $ ~ @ f % k # # & m-+3m.

RBt117 Ef$M%dB@BEffil'a7B:

I 4 % 32%: R&$ Hamilton gz+m-g Hamil ton gF 485

I $#gG Hamilton %&,&P

] 3-+ Hamilton s 4 , f i H J , JL '#$&I,&-+ Hamilton %f , U f i g R B Ha-

milton g$&~[~*~' , & ~ ~ ~ ' " 7 5 @ E ~ ~ % f & m Z % @ 4 t B @ . .Bill31 KdV %BI&FJShk 4' 3nT;XZ Hamilton B g

d j 2 d 1 H - + - - u p + - - u , . @Em J = K , L - - J - ~ H , f i J = ~ d x (i) 3 d x 3

5 JL = H $@,&A+ Hamilton %jY&El$ {f;) f&%f$m&Sq@@. %-%EM--+ Hamilton %ftifttlZ@@ekf3Z1J-EEil$A%c91. H&%kb%iiE Hamilton $$4$tJ&%E 3

i) %%m Hamilton $$4%$$4%f%fR%EilETS%%RGf Zm&X.

r m b @ a ~ ~ $R A o p @ ~ a ~ Mdf5ft$R% f r - @ ( f r , M ) ~2~ %tEIBlft1]8i'% Hami-

lton B4@%&fiB%&U1, #H%tifttlT#Ti&%FIf ~ % % m Hamilton $$=Em-+ I ~ R % B % i 5 1 . $ Z ~ ~ f i % B i ~ ~ ~ a ~ k y %f--EEEZm Hami1to.n B4?9FdT%T

#%B~$lJXIl%%~#HB@M$Sf;%kb2~&TS+ Hamilton BT. , ;P**B'-1FR%i%RB,Qi% I 2-+E&mtE$S%(quG!W&qZPW)Y

E =- {e;li€ I } ,

x SZ$EBB% c kEB E %t;:E$trX%J!k!Gr~ L c ( E ) , Z SZ&&%%k, Z: SZ*& ~ F & P B , R a Z * w a Y # ~ ~ ~ a ~ a $ u ~ w ~ z + ~ m m ~ + ~ ; i % s f i ~ a m ~ ~ 5 t * P W L r l E l $ Y ?awE%%m.

JllJrll ( 2 . 2 ) t i f f iaNB3HZ Hami l t on i$f3KJ%%%@%: ( a o b ) o c = ( a o c ) o b , a , b y c E X, ( 2 . 4 )

( a o b ) o c + c o ( a o b ) = ( c o b ) o a + a o ( c o b ) , a , b y c E X . ( 2 . 5 )

iG%t%%%Ri%%BT K = IIKiiII :

Re biim E C, i,i E I , m E Z l , # E H E S B j 2 K J V ( i , j ) E I X I , biim #YmRRRPF! 'i'+-%. #Yk%-i%i%%B3H%'%%&$k??$$3 K 2RKJ Hamil ton BRfllR: & 1 E H f A E B (2 .2 ) l@%N Hami l t on B 3 , K ~ E B (2 .6) &Xf8%t%&#%B

3,FIIJ H + K % Hami l t on B 3 K J % E % @ f A

-bijm LI ( - l ) *b j jm7 m 2 0 , ( 2 . 7 )

c o ( i , i y k ) + c o ( k , i , i ) = c o ( k Y i , i ) + c o ( i y k y i ) , ( 2 . 8 )

c l ( i y i y k ) = c d k y L i ) , (2.9)

~ z ( i , i , k ) + ~ 2 ( k , j , i ) E 0 , (2.10a)

c z ( i > i ~ k ) + @% ( i ~ i ~ k ) rn 0 , (2.10b)

c 3 ( i , i , k ) = c 3 ( k , i , i ) = c 3 ( i , i , k ) , ( 2 . 11 )

c , ( i , i , k ) = 0 , m 2 4 , ( 2 . 12 )

s*

dcf &&A% n + ~ = o ( c i ~ , ej,) -'C biirn~"', &k ( 2 . 14 ) 3.M-=F ( 2 . 7 ) 8. TK%B

m

(2.15) 6. lZl H l%J%@@%$l%

- --

4 &I 92%: Hrmilton A%!$m-& Hamilton 487

cz ( i , i , k ) + cZ(4, i , i ) -. 0,

2 c z ( i 7 i 7 4 ) - cZ( i , i , k ) + cz ( i , k , i ) + c2(k , i , i ) = 0 ,

ZE?%=+&&T p3. l p Z @EI%%E!Z3%%. XE%-&T%=&%Ek c z ( i , i 7 k ) + cz (k , i , i ) + c z ( i 7 k , i ) = 0 ,

T%%B$ (2.16) %B? (2.10).

3 m = 3 M y (2.16) RP% -c3( i , i , k ) l ( l + pI3 + c3(?,i7 k ) d i + P ) ~ + cd17 k7 i )p13

+ c 3 ( k 7 i 7 i ) ( l + p)A3 - c 3 ( k , i 7 i ) ( i + - c3( i7k7 i ) lp3 - 0,

%& 2'. LZpZ mEI%Bq%: c3( i7 i7k) = c 3 ( k 7 i 7 Q 7 c 3 ( i 7 i 7 k ) = c 3 ( i , i 7 k ) ,

&W p4. i 3 p . l p 3 mm%BBa%s. &&!$I (2.16) %fiT (2.11).

3 m > 4 i$jy%& i3pm-2 E i2pm-' @EJf#: c?cm( i , i7k ) - c?cm(17i7k) = 0 ,

cl"c,(i,i,k) - c?c,(i,i,k) = 0.

& c m 0 . 2 - 4 . . ifik (2.16) q f i ? (2.12).

gkEs (2.16) %fi? (2.8-2.12)7 B& (2.141, (2.15) %f i? (2.7-2.121. X ' (2 .14) , (2.15) $?AH + K% Hamilton ~3i$I%~&#"', &zR%&%ik%g. zq.

3 { c t ) X$%, RP c f l = c!i B$, EB (2.8) qf# co(i , i ,k) + c0( i , k , i ) = 2co(k , i , i ) ,

c o ( i , k y i ) + co(k , i , i ) = 2co( i7 i , k ) ,

@g@%qBfg ~ O ( ~ 7 j y k ) a ~ O ( k , ~ , i ) * ?% c 0 ( i 7 i 7 k ) = c o ( k 7 i , i ) = c0(j74, i ) ,

i&

c0(i,1,k) = co( i , k , i ) . (2.17)

&A,--EG (2.17) %G (2.81, Bhk (2 .8)@(2.17) .

EB,(2. 10a) f9$1J cz ( i , i , k ) = - c z ( i , k , i ) , i&EB (2. l o b ) FJ% c,(k,i , i) ;- 0 , BP cz( i , i ,k) = 0. (2.18)

&kY--R (2.18) @@$!I (2.10), & (2.10) (2.18).

gg311 (2.9) %I%? c,(i , i ,k) = c , ( i , k , i ) , (2.11) %f i? c3( i , i , k ) = c 3 ( i y k 7 i ) .

TERflIG #* i2 Hamilton g+ H = I I H ; ~ ( J %:

4 rn G ~ * f f : zPB,q Hamilton k;,g+w--& Hamilton gT 489

x ~ ! ~ b i ~ , = O , m 3 2 , m > 4 , I

3 { c $ } E8%7BP c?j a -cf i W7%4Ui&qH%%l: #* 2'" ig Hamilton $$F H = \\Hiill %:

g+ { c f i ) B-BEa%Nq4'iB2E17 K BE8 (2 .6 ) ;'ifXN%4 % % f[ F 7 NIJH+K%

Hamilton gljl .N%%%#~%~ (2.7) E&~L,H

s 3. *sfi@ W = Lc(D,) P*@J Hamilton

3~%8+2i$3t$~&1&-+$RB~ Nllt"H :Rq!j(I, PI)%%& E ~ % ~ E ~ N & ! J ~ ~ [ F S - - +

Hamilton ST, EB $k?&{iI&B& -+$I%% W % R I ~ % F ~ l - - ~ @ E ~ Hamilton $$Fa "

i2 Q ( x , y ) % X Y Y NzZ&?B&R%~ q ( x ) = x q,xr, n > 0. $E Q ( x > y ) 1% r=o

xx%lE%T [ 9 1 fm-F:

[ f rgl = g a f - f a g . (3.3)

% 4% f% PIJ ~ [ f , g i , h i - f . a 2 g . h - a2f . g . h - - f W a g . a h + a f . g . a h .

& lkWf€iiEBa: [ [ f 7 g l y h l + [ [ g , h I , f l + [ [ h , f l , g l = 0 ,

MK Q ( x y y ) $ZB [ * , * I T&&++$RB. 22 I q 3 ! J y @ 1 ZZO+y .$ e i ~ .t.'yAY i , n € Z i 7 WIJ w = L d D J = Q(x,y)a i%l

[e;A,eirl = q ( ~ ) y ( x i y " xilyi-' - xiy" xi py"-')

= q ( x ) ( 1 -

& Hamilton $$+ H - JIHiiIl %

: + i + n d

~ i i = x q k - i - i ( u ~ ) + u ~ - - ) , i y j t 1 . q r ~ C y ~ < r < n t = i + j d x

I Hamilton

( 2 ) i3 1 a { O ~ l , - - - y m ) y m 2 0 , + eil %7;9J-~%BB532%3

4 #&I, 552%: R?B@ Hamilton %;\51E@m--% Hamilton 49 1

- J {C qi [CjU:) + (ci% + C ~ U O - 5 1.1 d x 1 I d s

I B # A ~ I J Hamilton $@=I-. H - (\Hiill:

d Hij ' cju!" + (c jui + ciui) -3 i , j E 1, d x I

%f'Td&!Xlt8%?% 1 . 1 FJG!GFR&FJZPR.@ c!, = 6kiciy i . i .4 E 1 . ci E C . i E 1,

f lu& (2 .3 ) %5Z!XlZB. %E (2 .4 ) &I (2 .5 ) . TE%#g.4 H - IIHijll:

Hij - [ C B U P ' + (c!, + c ~ ) u ~ -- . i . i E I . d l d x t En

d Hii a ciu;" + (cjui + ciui) -9 i y 1 E I d x

(3.6)

Hamilton $$IF,

%1%3RPt117 R: 8B 3 %I%?#&BF H = IIHiiII :

d Hij cjujl) + (cjui + ciuj) -- 7 i , j E I . ci E Cy i E I d x

E Hamilton $$T.j&BlPP%% I fE&.

lo , k > i + ~ t m . (3.7)

T;~i-f-B%IE!% Hamilton BaBF H = \JHiil(. El

#@jg!J Hamilton BF H a \\Hiill:

,g-+ Hamilton $$IF, 1St;ggjf$)-T a - 1 E m = 0 &%35ti8R%'".

4 m 32%: xP& Hamilton %a+&)--% Hamilton 493

§ 4. -+=~&#J~@&@EB Hamilton $3.

'ei X ej = x b$ek , ei* ei = x d?;ek, i , j € I , . . . . . .

(4 .10) 4 .,, a . . 4 : ,

All (4.7-4.9) %fiT(RB%& d!; = -d$) . . . . . .

. . ( a X b ) X c = 0 ,

( a X b ) * c + + . ( b * c ) X a + ( c * a ) X b = 0 , . .

(.*b).*.c+ ( b * c ) * a ~ + ( c * a ) * b = O ,

2% a , b y c € X. X { T ? j } S I B & , & X qFsy S&zB x@.xi&IJfiY-+.=&q2#$

+fi%fr[6J. . .

--FZ$Uk% { q 3 ? ; ( l y p ) } %gg&mBg&%&@Rg%g-F,'.: my: .:>;, . . ;.:,,:>.. ;.

(?%$ {b?i} S {d? j } B3Jff%$By .EL { ~ ! ~ } , ~ F S ) X J Hamilton B 3 & 3 % &#E& (4.10) SZX&I+IZ%~KZB x $n * %E-FPl&#:

( 1 ) 52% x @ ~ & f i - ' P = . & " S 2 & 4 R & ~ ; . . : . . . ' . , ' , ( 2 ) ti* @x&,E$,-+$R~;

..l ;.* <i..

( 3 ) ZB* $n* G#%: . . i j .. !: * . . : , . ( i . . . .

( a X b ) * c + ( b * c ) X a + ( c * a - ) X - b = : O , a , 6 , c E X, (4 .12 )

E--+ Hamilton B3, . . . . . . . .;! P : . . . <.:; < .. . ; . . , . , . I'. .

, > . . , * .

4x4 4x3: Hamilton Z%$($J-% Hamilton gF 49 5

CI$ Bi+i.p+k, i , j , k E 1 .

( 2 . 1 9 ) ~ijSf8E~@,!i?m Hamilton $$+ H L Q :

K = C B m D m B m ~ ' ( b i i m ) ( p + l ) r ( p + l ) , b i i m = ( - l ) m ' i b j i m , i , j E I , m 2 0 , ( 5 . 3 ) m>O

< . El K E%, &S Hamilton $$+.

XY H + K Hamilton ~32fllGT9I%ik: $ 1 6 $4: H , K f * 5 l h f & " ( 5 . 2 ) , ' ( 5 . 3 ) $&&I Hamilton $$+, 211 H + K %

I I

Ht-lamilton $~%XJ~ES&~+E ' I

B r n - 0 , r n # i , 3 , x . - 4 5 . 4 )

Ti. m = 0 , 1 , 3 , &I ( 5 . 1 ) %%1kW ( 2 . 2 0 ) , Wl]%

b,,, = bl,, 3 31 E I @ r + s = p + 1 l$, (5.8)

%iE (5.7). Ti2 O<r + s < p - 1. $E (5.6) $, @ i = r + s + 1( < p ) , j =

P - s - l ( > O ) , 4 - $ 9 Wq i + k 3 p - 1 ( ~ 3 & ( 5 . 6 ) S & 2 ~ % , m j & ~ = b r I m y

K l f i brsm = 0.

B i E ( 5 . 8 ) . Ti9 r + s - - p + l , l E 1 . a ( 5 . 6 ) $ , @ i = p , j - r , q = s , WIJ \

brsm = blpm, + blm blpm &Pf+iE. E % y + c z i i ~ b i ~ , c2ji2 = biz, 0 < i < P Y Bll@iEZ3%.2&$Zf&. iE%.

t8 k~b3%lfilFJWla% BLl ( d + 2 s D ) + 2 b.Dm PI&1%3$&lir% m= 0

(ub + 2uDD) + blD + b3D3

31 Hamilton gF.3.. %ZF&[7,8l$N Hamilton 1%3.%BeB 6 N%@J.

4 * m [ 1 1 I?. M. reJIb@aHn, I?. fl. f l o f i ~ a ~ , ~ ~ M H J I ~ T O H O B ~ I O n e p a T O p h l H CBRSaHHble C H H M H m r e 6 p a H s e c ~ ~ e

CTPYKTYPM, @ Y H ~ . OHOAU3, 13:4 (1979). 13-20. [ 2 ] F. Magri , A Simple Model of the Integrable Hamiltonian Equation. J . M a t h . P h y s . , 19:5 (1978),

1156--1162. 1 3 3 P. J . Olver, Applications of Lie Groups to Differential Equations. Springer-Verlag. Berlin

1986. [ 4 ] G. 2. T u , The trace identity, a powerful tool ' fo r constructing the Hamiltonian structure of

integrable systems. J . M a t h . P h y s . , 30:2 (1989). 330-338.

[ 5 ] M. M. reJIbL$aHn, I?. fl. n O p @ M a ~ , ~ ~ M H J I ~ T O H O B M O n e p n O p b l U ~ ~ C K O H ~ V H O M ~ D H ~ I ~ a J I r e 6 D b l n H ,

15: 3 (1981). 23-40.

C 6 I ~ Z R Y R % ~ % % % ~ ~ ~ W ~ % Y ~ E # % E ~ % & E % ~ B Y 3 : 2 (19861, 37--48. 4 [ 7 ] G. 2. T u , A Simple approach to Hamiltonian structures of Soliton equations. -I., I I Nuova

C i m e n t o , 73B: 1(1983), 15-26.

[ 8 ] M. Boiti, C. Laddomada, F. Pempinell i & G. 2. T u , On a New Hierarchy of Hamiltonian So- l i ton Equations, J . M a t h . P h y s . , 24:s (1983), 2035-2041. 1

[ g l %ZR,--EW%'Z ~ a m i l t o n w ~ ~ ~ E z % ~ # ~ % Y k & 2 B k Y % t W , 2 3 : 1 (1989), 9 -18 ,

Eklk~$BRillTBsl&% im ( i + 2 r D ) + b.Dm LhBJtS)PTF$f Bb m= 0

(ub + 2uoD) + b,D + b3D3

31 Hamilton SF. %*i%[7,8 I$!% Hamilton SF%@2B 6 FB%@J.

* * a [ 1 I ki. M. ren@a~n, ki. R. f i o p @ ~ a ~ , r a ~ u n b ~ o ~ o a b l o n e p a T o p H H c e f i 3 a H H H e c H H M H a n r e 6 p a n q e c u u a

C T P Y K T Y P ~ I , @YHKY. UHUAU3, 13:4 (1979). 13-20. [ 2 ] F. Magri , A Simple Model of the Integrable Hamil tonian Equation, J. M o t h . Phyr . , 19:5 (1978),

1156-1162. [ 3 ] P. J . Olver , Applications of Lie Groups to Different ial Equations, Springer-Verlag, B e r l l t l

1986. [ 4 1 G. Z . T u , T h e trace identi ty, a powerful tool ' fo r construct ing the Hamiltoninn clrclct l l ra 111

integrable systems. J. M o t h . Phyr . , 30:2 (1989). 330-338. I [ 5 ] ki. M. r e n b @ a H n , ki. R. AOP@M~H, ~ ~ M U J I ~ T O H O B ~ I O n e p a T O p b l U ~ ~ C K O H ~ ~ H O M ~ P I I M O lI~lI'tlfil)ld JI#, I

15: 3 (1981). 23-40.

[ 6 I ~ ~ ~ % Y R % @ ~ % ~ ~ F ; ~ : W ~ % I ) ~ B ~ ~ W E R % & E ~ ~ ~ 3 : 2 ( l 9 8 6 ) , 37--48, 1 [ 7 ] G. Z. T u , A Simple approach to Hamil tonian s t ruc tures o f , Soliton equrt ionl . - I , , I 1 Nurrvo

Cimenzo, 73B: 1(1983), 15-26.

[ 81 M. Boiti, C. Laddomada, F. Pempinel l i & G. Z. T u , O n a New Hiernrahy H n m i l t o ~ ~ i ~ ~ t ~ So- l i ton Equations, J . M o t h . Phyr . , 24:8 (1983), 2035-2041.

[ 9 I 4fcRY--Egg3 ~ a m i l t o n %#&I&!J%E~&E#X.~%, k$52j!3kqqjM, 28 : 1 (1989), 9.- 14,