Pergamon Appl. Math. Lett. Vol. 11, No. 3, pp. 71-74, 1998

Copyright©1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0893-9659/98 $19.00 + 0.00 PII: S0893-9659(98)00035-4

Integrability of the Hdnon-Heiles System

R. G. SMIRNOV Department of Mathematics and Statistics, Queen's University

Kingston, Ontario, Canada, K7L 3N6 smirnovr~mast, queensu, ca

(Received and accepted June 1997)

Communicated by B. Fuchssteiner

A b s t r a c t - - A n integrable case of the Hdnon-Heiles system is isolated by deriving a suitable bi- Hamiltonian structure leading to its complete integrability in the sense of Arnol'd-Liouville.

K e y w o r d s - - B i - H a m i l t o n i a n dynamical systems, The Hdnon-Heiles system, Complete integrabil- ity.

1. I N T R O D U C T I O N

The approach to integrability through bi-Hamiltonian structures has proven to be quite powerful in studying of both finite- and infinite-dimensional evolutions equations (see [1,2] and the relevant references therein). In the finite-dimensional case, it rests on the following facts. Given a general Hamiltonian system: (M,P, XH), where (M,P) is a Poisson manifold, XH-- the correspond- ing Hamiltonian vector field uniquely defined by the Poisson bivector P and the Hamiltonian function H (total energy) in the following way:

XH = [P, HI. (1)

Note that throughout this paper the bracket [, ] is that of Schouten [3]. Then if the vector field XH has two distinct Hamiltonian representations:

XH1,H2 -~ [Pl , H1] --- [P2, U2] (2)

(it implies that either pair (P1,H1),(P2, H2) may coincide with the initial one: (P,H)) , the quadruple (M, t)1, P2, XHI,H2) is called bi-Hamiltonian system. In certain cases, integrability in the sense of Arnol'd-Liouville of its bi-Hamiltonian vector field XH1,H 2 is assured if the Poisson bivectors P1 and P2 satisfy the compatibility condition:

[P1, P2] = 0. (3)

See [4,5] for details. In particular, a bi-Hamiltonian dynamical system (2) defined by a pair of compatible Poisson bivectors having two degrees of freedom is integrable if the two Hamiltoni- arts H1 and/ /2 are functionally independent.

The author acknowledges with gratitude a useful discussion on the subject of this work with O. Bogoyavlenskii. The research was supported in part by the National Science and Engineering Research Council of Canada.

Typeset by .AAd~TEX

71

72 R.G. SMIRNOV

The guiding concept of this approach to integrability is the existence of a compatible Poisson pair defining system (1). Therefore it is natural to seek a bi-Hamiltonian representation with a compatible Poisson pair for a given Hamiltonian system to study its integrability. Although in actual applications, it is usually quite a nontrivial problem. In some instances, it is possible to transform a given Hamiltonian system into a bi-Hamiltonian form by employing a suitable master locally Hamiltonian (MLS) vector field (see [6]), namely such a vector field Yp, whose commutator with XH : [Yp,XH] is a locally Hamiltonian vector field with respect to P : [[Yp,XH], P] = O, while Yp itself is not: [Yp, P] ~ O. Here the operator [Z, .] means the Lie derivation along a vector field Z in the space of contravariant tensors on M. We note that the definition of a MLH vector field mimics the definition of a master symmetry introduced by Fuchssteiner in [7], which in sequel has become an important component part of the bi-Hamiltonian theory. However, this approach has its drawbacks: it is not always easy to derive an appropriate MLH vector field generating a nontrivial new Poisson bivectors defining the corresponding system. All of these problems can be avoided if one looks for a bi-Hamiltonian representation defined by a pair of constant Poisson bivectors. We shall show that such a situation is possible for a Hamiltonian system with two degrees of freedom, moreover, it can lead to complete integrability of the latter.

2. T H E H ] ~ N O N - H E I L E S S Y S T E M

Let us consider as an example of a Hamiltonian system with two degrees of freedom the H~non- Heiles system in its general form, i.e., defined by the following vector field XH:

0 + p20~ 2 + ( -Cql + Bq~ - Aq 2) ~Pl - (Dp2 + 2Aqlq2) (4) XH = Pl ~ql

with the Hamiltonian (total energy) H:

1 1 3 H = ~ (p2 + p2 + Cq2 + Dq2) + Aqlq22 _ .~Bql" (5)

The corresponding Poisson bivector in this case is canonical:

0 0 0 0 P = ~ql A ~pl + ~q~q2 A Op---~. (6)

Although in general, this system has a rather irregular behavior, there are known three integrable cases defined by certain interrelation between the constants A, B, C, and D:

A I. = - 1 , c = D , (7)

A 1 II. ~ = - ~ , C, D arbitrary, (8)

A 1 III. B = - 1--6' D = 16C. (9)

They have been isolated by using the Peinlev~ method (see, for example [8]). Alternatively, Fordy [9] employed the technique connecting the Hamiltonian formalisms of stationary and non- stationary flows to prove that these were the only three integrable cases of system (4). We shall solve the following problem: by using the bi-Hamiltonian method, to integrate explicitly system (4), i.e., to derive an integrable case.

To do so, we need to present a second Hamiltonian formulation for the vector field (4), which will give us the second first integral necessary for complete integrability.

I. Let us find first a second Poisson bivector, in the class of constant Poisson bivectors, preserved by XH : {Pc, [XH, Pc] = 0, Pc - const}. A general constant Poisson bivector in

The Hdnon-Heiles System 73

our case (i.e., just a skew-symmetric two-contravariant tensor with constant coefficients) takes the form:

(9 a b.__O.0 a a O d --~-~ A O a O ~Pl O P c = a - ~ q l A ~ q 2 + Oql A ~ p l + C ~ q l A ~ p 2 + (9q2 ~ p l + e ~ q ~ A 0 - - ~ + f _ A ( g p 2 "

The condition [XH, Pc] = 0 yields

0 = [XH, Pc] = (d - c)-~Ta~ A + 2aAq2 A + ( - f + aD + 2aAq,) A

O~2 0 2aAq2 0~2 A O × ( - f - aC + 2aBql) A ~Pl Op2

+ (cC - 2cBql + 2eAq2 - 2Aq2 - dC - 2dAql) A Op2'

which immediately entails the conditions on the constants a, b, c, d, e, f , and A, B:

A a = f = 0, b = e, d = c, B -1 .

Therefore, a general Poisson bivector Pc with constant coefficients preserved by the vector field (4) enjoys the form

Pc --= a 0 0 ~ql 0 ~_.0__0 A 0 5 0 0 (10) -O~ql A-O-~pl + ~) A-~-2p2 -1- Oq2 -~pl + -~2q2 A op----2"

II. We want a Poisson bivector of the type (10) to provide the Hamiltonian vector field (4) with a Hamiltonian representation. Suppose that for a function//2 and a Poisson bivector (10) such a representation exists: XH = [Pc,/-/2] or X i = P~iH2,k, i = 1 , . . . , 4. Then, if in the position-momenta coordinates (ql, q2, Pl, P2), the differential of/-/2 is given by the vector (hi, h2,h3, h4) T, we have (see (4)):

Pl ---- ahl + bh2,

P2 --- bhl + 5h2,

which in general is impossible, since Pl and P2 are linearly independent as coordinates. Hence, we arrive at one of the following two cases.

(1) 5 # 0 , b = 0 , or (2) a = 0 ,

Case 1 will not give a new functionally independent first integral, because the invariant Poisson bivectors in this case are multiples of the canonical Poisson bivector (6). Consider Case 2:

( 0 0 0 0 ) P 2 = b ~--~qiA~-~p2+~-~q2A~pl . (11)

Then the differential of/-/2 in the coordinates (ql, q2, Pl, P2) is given by the vector

1 dH2 = -~ (Dq2 + 2Aqlq2, Cql - Bq 2 + Aq2,p2,pl) T

Now we easily derive the explicit formula for the Hamiltonian H2:

1-12 = Cqlq2 + Aq2q2 + 1Aq~ (12) 3

where C = D. Therefore, the dynamical system of the vector field (4) is bi-Hamiltonian and completely integrable as such for A / B = -1 , C = D having the two first integrals (5) and (12) corresponding to the Poisson bivectors (6) and (11). This is exactly the integrable Case I, (7).

74 R.G. SMmNOV

3. C O N C L U D I N G R E M A R K S

In this letter, we have integrated the general Hdnon-Heiles system by deriving a proper bi-Hamiltonian representation for it. The obtained second first integral together with the initial Hamiltonian correspond to the integrable Case I.

We note that Caboz et al. in [10] studied the inverse problem: for an integrable case (Case II, (8)) of the Hdnon-Heiles system, there was presented an explicit bi-Hamiltonian structure by making use of the method of separation of variables.

R E F E R E N C E S

1. I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley & Sons, New York, (1993).

2. P.J. Olver, Applications of Lie Groups to Differential Equations, Second edition, Springer-Verlag, New York, (1993).

3. J.A. Schouten, Uber differentalkomitanten zweier kontravarianter GrSssen, Proc. Kon. Ned. Akad. Amsterdam 43, 449-452 (1940).

4. F. Magri, A simple model of the integrable Hamiltonian equation, d. Math. Phys. 19, 1156-1162 (1978).

5. I.M. Gel'fand and I. Dorfman, Hamiltonian operators and algebraic structures related to them, b~unet. Anal. Appl. 13, 248-262 (1979).

6. R.G. Smirnov, Bi-Hamiltonian formalism: A constructive approach, Lett. Math. Phys. 41, 333-347 (1997).

7. B. Fuchssteiner, Mastersymmetries and higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Prog. Theor. Phys. 70, 1508-1522 (1983).

8. B. Grammaticos, B. Dorizzi and R. Padjen, Peinlevd property and integrals of motions for the Hdnon-I-leiles system, Phys. Left. 89 (A), 111-113 (1982).

9. A. Fordy, The Hdnon-Heiles system revisited, Physiea D 52, 204-210 (1991). 10. B.. Caboz, V. Ravoson and L. Gavrilov, Bi-Hamiltonian structure of an integrable Hdnon-Heiles

system, J. Phys. A: Math. Gen. 24, L523-L525 (1991).