When do recursion operators generate new conservation laws?

Physics Letters B 277 ( 1992 ) 137-140 North-Holland

PHYSICS LETTERS B

When do recursion operators generate new conservation laws? Gmseppe Marmo a and Gaetano Vilasl b " Dtpammento dl Sctenze Ftstche, Umverstta dt Napoh. 1-80125 Naples, Italy b Dtparttmento dz Ftstca Teonca e SMSA. Umverstt& dl Salerno, 1-84100 Salerno, Italy and Istztuto Naztonale dt Ftstca Nucleate, Seztone dz Napoh, 1-80125 Naples, Italy

Received 18 October 1991

Two hamlltoman structures for the Kepler problem are constructed However, the recursxon operator Tdoes not lead to a new functionally independent hamdtonmn The result holds for any vector field satisfying the energy-period theorem hypotheses, and admitting a tensor field Twhlch factonzes via two symplect~cstructures

1. Introduction In recent years there has been a renewed interest m completely integrable h a m l l t o n i a n systems, specially m conjunction with the study o f mtegrable q u a n t u m field theory, Y a n g - B a x t e r algebras and, m o r e recently, q u a n t u m groups A m o n g &fferent m e t h o d s d e v e l o p e d m the analysis o f these systems, the inverse scattering transform ( I S T ) , based on Lax representation, is universally recognized as an I m p o r t a n t integration algorithm Nevertheless, a priori criteria o f lntegrablhty have been estabhshed only by m e t h o d s m o r e &rectly related to group theory [ 1,2] a n d to a familiar procedure in classical mechanics, looking at sohton equations as d y n a m i c a l systems on a (infinited i m e n s i o n a l ) phase m a n i f o l d [ 3 - 8 ] This p o i n t o f view was also suggested b y the occurrence in the IST o f a peculiar operator, the so called recurston o p e r a t o r [ 9 ], relevant for the effectiveness o f the m e t h o d , which naturally fits in this geometrical setting as a m i x e d tensor field on the phase m a n ffold M The belief is w i d e s p r e a d that once a recurslon operator has been f o u n d all conserved functmnals for the system can be constructed starting with the given h a m l l t o n l a n function In this paper, it will be shown Supported m part by the Italian Mmlstero della Umverslta e della Rlcerca Sclentlfica e Tecnologlca

tn an example that the existence o f a recursxon operator does not give rise to additional conservation laws even for completely mtegrable h a m l l t o n l a n systems This peculiar result o n . n a r e s from the energy-period t h e o r e m ( E - P ) [ 10,11 ] We show that for any h a m l l t o m a n system with periodic orbits a n d p e r i o d satisfying the hypotheses o f the E - P theorem, a recursion o p e r a t o r will not take the h a m l l t o n i a n function into a functionally i n d e p e n d e n t one The p a p e r is organized as follows First, we fix notation a n d recall an alternative f o r m u l a t i o n o f complete integrabfllty After the construction o f the recursion o p e r a t o r for the K e p l e r problem, a very short account o f results on the relation between p e n o d a n d energy m periodical d y n a m i c a l systems is given Therefore we deal with a recursion operator and show our m a i n result

2. Complete integrability and reeursion operators Complete integrablhty o f hamlltonlan systems with finitely m a n y degrees o f f r e e d o m is exhaustively characterized by the L l o u v d l e - A r n o l d t h e o r e m [11,12] A n alternative characterization which m a y apply also to systems wtth infinitely m a n y degrees o f freed o m can be given as follows Let M denote a smooth dlfferentlable manifold, X ( M ) a n d A ( M ) vector a n d covector fields on M

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137

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PHYSICSLETTERSB

With any (1, 1 ) tensor field T on M, two endomorphxsms T ^ X(M)--,X(M) and T v A ( M ) ~ A (M) are associated

27 February 1992

_ Ip 0 Po 0 Po 0 A= m1 for+ r-~ ~ + r2sln20 00 l(p 2

---f3

T(a, X) = (or, T ^ X ) = ( Trot, X), VX~X(M), oteA (M) The N1jenhuls tensor [ 13 ], or torsion, of T is the ( 1, 2) tensor field defined by

p~ ~ O

p~cos0 0

k 0]

°+s~n20"]~Pr r2slnaOOp° r'EOpr(3)

It is globally hamlltoman with respect to the symplecUc form

co= E dp, Adq,,

(4)

t

Nr( ot, X, Y) = ( a, Hr( X, Y) >,

(1) with the hamlltonlan H given by

where X, Y e X ( M ) , c ~ A ( M ) Hr(X, Y) Is given by

and the vector field

Hr(X, Y ) = [ ( ~ r ^ x T ) ^ - T ^ ( ~ x T ) ^ ] Y ,

(2)

£ax denoting the Lm derivative with respect to X The following theorem is proven in ref [ 3 ]

Theorem A dynamical vector field which admits an lnvanant, mixed, dlagonahzable tensor field Twith vamshlng N1jenhms torsion and doubly degenerate elgenvalues 2 without stationary points (d2 # 0) is separable mtegrable and hamlltonlan, 1 e , a separable completely lntegrable hamlltoman system The proof is given observing that N r = 0 implies the mtegrablllty, in the Frobenlus sense, of elgenspaces of T LP~T= 0 lmphes the separability ofd along the elgenmamfolds, m dynamics with one degree of freedom, each of them with a constant of motion We notice m passing that .~e~T=0 provide us with a Lax pair [14] A ( 1, 1 ) tensor field of this kind acts as "recurslon operator" [ 6-8 ], 1 e , it can be lteratlvely applied to A to get symmetries dk= ( T ^ )kA and also constants of motion Hk by dHk= ( T v )kdH One showsthat [ ( T ^ ) k d , (T^)k+rA] = 0 a n d {Hk, Hk+r}=O [3,15]

1

2+

V(r) = -

-

k

(5)

r

Let us introduce angle-action coordinates (J, 0) The map, which is not a dlffeomorphlsm, between (J, 0) and (p, q) is constructed by

/

Jr = J , = - - ~/p2O+ -P~ Sln20

.f2mk +mgk r

pg

r2

r2sln20

_p~)/

J, =J3 = P , , 1

0, = - (jr+Jo+j,)2 [ -m2k2r2 + 2mk(Jr + Jo + j,)2r_ (jr + jo + j~)2(Jo + j,)2] 1/2 m k r - (Jr + Jo + Jo) z

+ arcsm

(Jr +J0 +J,)X/(L +do +j~)2_ (Jo +j~)2, 02 = 0, - aresm

[mkr- (Jo +Jo) 2] (Jr + Jo + J,) , / ( Jr + Jo + JD2( Jo + Jo) 2

(Jo + Jo)cos O , x/(Jo+J¢)2-Jg

3. A recursion operator for the Kepler potential

03 = 02 + arcsln

138

pg

Jo=J2= X/p ~o+ s~n20 P~ -P~,

-arcsln

The vector field for the Kepler problem, in spherical-polar coordinates, for R3 - {0}, is given by [ 16 ]

p2

+

J, cotg 0

. _

4- o +jo)2- Jg

+0

The Kepler hamlltoman H, the symplecnc form co and the vector field J, in these coordinates, become

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PHYSICS LETTERSB

nates where the tensor field T is simply written as

mk 2 (jr+Jo+j~) 2 ,

H=

(6)

to= ~, dJhAdOh,

~ + (S+)hkd0h® O~-k) 0 T = ~hk( ShkdJh®O

(18)

(7)

h

Moreover

A= (jr.l_jo.~j¢)3 ~1 dl- -~2 + ~ 3

(8)

It is easy to see that the vector field A is globally hamlltonlan also with respect to the symplectic form

tot to' = ~ Shk dJh ^ dq~k,

(9)

hk

with hamlltonlan H ' given by

2mk z Jr+Jo+Jo'

H'=

27 February 1992

1{| J1 2\J3-J2

J2 Jl+J3 ./2

-~

dH

(19)

Thus the iterated application of T does not produce new functionally independent constants of the motion We are going to show now that this particular prevails for periodic systems when the period is a smooth funcUon of the initial condition

(10)

4. Relation between period and energy in periodical dynamical systems

and where the matrix S is given by

s==lJ2-J3

T v dH=k

J3 ) J3 Ji +J2

(ll)

Remark The matrix S cannot be identified as a transformation jacobian as it is clear from the fact that the S doffs are not closed one-forms In the onginal coordinates (p, q) the symplectic form to' ~s simply written as to'= ~ dK,^da,,

l

(12)

where the functions K,(p, q) and ot,(p, q), defined by Kl = 1 [J12+ ( J E - J 3 ) 21 (P, q),

(13)

K2 = ½J2(J1 +J3) (P, q),

(14)

K3=lJ3(Yl +J2)(P, q),

(15)

a,=q),(p,q),

(16)

are considered as functions of p, q by means of the map J, = J, (p, q), 0, = q~,(p, q) previously defined As a consequence a mixed invariant tensor field T, defined for not degenerate to, by

to(TAX, Y)=to'(X, Y ) ,

(17)

can be constructed The vanishing of the N1kenhuls torsion and the double degeneracy of the eIgenvalues of T is more easily checked, however, in the angle-action coordl-

Let qb= {0t} be a one-parameter group acting on a manifold M, and A the infinitesimal generator of It is supposed that ~b has a period, i e , that a posture function of class C ~,P xeM---,P(x) eR exists such that ~etx) = x for all x m M The following theorem can be proven [ 10 ]

Theorem IfA is a globally hamlltoman vector field and H the corresponding hamlltonlan, P and H satlsfy dP ^ d H = 0, m a neighborhood of any point x where P exists From here it follows that P is a function of H A rough idea of the argument used in the proof can be provided by the following consideration Using extended coordinates (p, q, t), we have that the twoform I2=Y~, dp, A d q , - d H ^ d t is preserved by the hamlltonlan flow ~a {(P, q), t}--,{Cl)a(p, q), t+a} If a=P(p, q) we get q~e {(P, q), t}--,{(p, q), t+e(p, q) } because of the periodicity Therefore ~,g-2--_g2~ dHAdP=0 It is now clear that all various alternative hamiltoman descriptions that we may build, via a recurslon operator T will satisfy alP^ ( T V ) k d h = 0 , l e , if dP¢0, (TV)kdH^ (TV)*+rdH=0 In this finite-dimensional setting it Is true, however, that {Tr(TA) k, T r ( T A ) h } = 0 , and T r T ^, T r ( T i, )2, T r ( T ^ )3 are functionally independent 139

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[3 ] I n mfimte dimension, however, 1s not easy to make sense of the trace of an e n d o m o r p h l s m

5. Conclusion S u m m i n g up, we have shown that the keplerlan vector field A admits two h a m l l t o m a n descnpUons tato=dH,

taro' = d H '

and that the recurslon operator T, constructed out of to a n d to', does not lead to a new funcUonally rodep e n d e n t h a m d t o n m n The result holds for any vector field satisfying the energy-period theorem hypotheses a n d admitting a tensor field T which factorizes m two symplectlc structures We find then that the existence of a n m v a n a n t ( l, 1 ) tensor field, connecting two h a m f l t o n m n descriptions of the same dynamical evolution, by no means imply the posslblhty of building new conservation laws out of the starting h a m d t o n m n From the nature of the proof it seems plausible that a similar theorem should hold true also in infinite dimensions Therefore our remark shows that much more care is needed when various ways of dealing with complete lntegrabfllty and conservaUon laws are considered to be eqmvalent

References [1 ] R Schmldt, Infimte dlmenstonal hamdtoman systems (Blbhopohs, Naples, 1987), and referencesthereto

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27 February 1992

[2] L D Faddeev and L A Takhtajan, Hamdtonmn methods m the theory of sohtons (Spnnger, Berlin, 1987), and references thereto [ 3 ] S De Flhppo, G Marmo, M Salerno and G Vllasl,Nuovo Omento 83 B (1984) 97 [4] G Vllasl, Phys Lett B94 (1980)195, G Marmo, Geometry and Physics, ed M Modugno (Pltagora, Florence, 1982) p 257, S De Flllppo, M Salerno and G Vflasl, Lett Math Phys 9 (1985) 85 [5] I M Gel'fand and I Ya Dorfman, Funct Anal 14 (1980) 71 [6]F Magrl, J Math Phys 18 (1978)1156 [7]B Fuchsstemer, Prog Theor Phys 68 (1982) 1082 [8 ] V E Zakharov and B G Konopolchenko,Commun Math Phys 94 (1984) 483, Y Kosmann-Schwarzbach, Geom6tne des systems blhamdtoman, Pub IRMA, 2, l (Lille, 1986) [9]PD Lax, Commun Pure Appl Math 21 (1968)467, 28 (1975) 141, Slam Rev 18 (1976) 351 [ 10] W B Gordon, J Math Mechamcs 19 (1969) 111 [ 11 ] R Abraham and J E Marsden, Foundations of mechamcs (Benjamin, New York, 1978) [ 12] V I Arnold, Les m6thodes math6matlquesde la m6camque classlque (Mlr, Moscow, 1976) [13] A FrollcherandA Nuenhuls, Indag Math 23 (1956) 338, A Nijenhuls, Indag Math 49 (1987) [14] S De Fihppo, G Marmo and G Vdast, Phys Lett B 117 (1982) 418 [ 15] S De Flhppo, G Marmo, M Salerno and G Vllasl, On the phase manifoldgeometryoflntegrable nonlinearfield theory (IFUSA, Salerno, 1982) [16]L Landau and E Llfchltz, M6canlque (Mlr, Moscow, 1966)