PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 63 (77), 1998, 143–146

INTEGRATION OF THE PERTURBATED JACOBI PROBLEM

Vladimir Dragović

Communicated by Stevan Pilipović

Abstract. It is shown that recently obtained family of integrable potential perturbations of the Jacobi problem for the geodesics on the ellipsoid can be integrated by separation of variables in the elliptic coordinates. The way of reduction to the Liouville case is demonstrated and the complete integral of the Hamilton–Jacobi equation is given.

1. Introduction

In the theory of the Hamiltonian systems, the completely integrable case has a very important position. According to Lionville’s theorem (see [1, 2]), the motion associated to the system has regular behavior. Unfortunately the integration procedure given by this theorem is ineffective. There are several methods of effectivisation of this procedure, classical and modern, as presented in the encyclopedic reviews [1, 2].

The integrability of the Hamiltonian system describing a particle moving under inertia on an ellipsoid

\frac{x^{2}}{a} + \frac{y^{2}}{b} + \frac{z^{2}}{c} = 1  (1)

was proven by Jacobi by separation of variables in the elliptic coordinates (see, for example, [1, 2]).

A family of integrable potential perturbations of the above system was recently obtained (see [5]) using Kozlov’s method (see [3, 4]). It was found in the form of Laurent polynomials:

V (x, y, z) = ∑_{r≥n,m,p≥s} a_{m,n,p}(a, b, c)x^{m}y^{n}z^{p}, s, r ∈ Z.

AMS Subject Classification (1991): Primary 58F05 Supported by Ministry of Science and Technology of Serbia, Project 04M03

The basic potentials V_{m_{0}}, m_{0} = −2t, t ∈ N, were given (see Theorem 1 in [5]) by the formulae

a_{m_{0}+2k,2s,−m_{0}−2−2(k+s)} = \binom{s + k − 1}{k}\frac{c^{k+s}(c − a)^{s}(c − b)^{k} ∏_{i=k+1}^{k+s}(m_{0} + 2i)}{b^{k}a^{s}(b − a)^{k+s}(−2)^{s}s!} (2)

where k ≤ s, and m_{0} + 2(k + s) < 0.

In this paper we want to show that these systems are integrable by the method of separation of variables in the elliptic coordinate system. On the example of V_{−4} we demonstrate how such system can be reduced to the Liouville case (see [6, 7]) and give the complete integral of the Hamilton–Jacobi equation. (The exapmle V_{−2} = 1/x^{2} has been studied by Kozlov in [3].)

2. The elliptic coordinates and the separation of the variables

If 0 < a < b < c are distinct positive real numbers, Jacobi’s elliptic coordinates ( in R^{3}) λ_{1}, λ_{2}, λ_{3} can be defined by the function which the triple (x, y, z) ∈ R^{3} maps the roots (λ_{1}, λ_{2}, λ_{3}), λ_{3} < λ_{2} < λ_{1}, of the equation

f(λ) = \frac{x^{2}}{a − λ} + \frac{y^{2}}{b − λ} + \frac{z^{2}}{c − λ} = 1.

Conversely (see [1]):

\begin{aligned}x^{2} & = \frac{(a − λ_{1})(a − λ_{2})(a − λ_{3})}{(a − b)(a − c)}, \\ y^{2} & = \frac{(b − λ_{1})(b − λ_{2})(b − λ_{3})}{(b − a)(b − c)}, \\ z^{2} & = \frac{(c − λ_{1})(c − λ_{2})(c − λ_{3})}{(c − a)(c − b)} \end{aligned}  (3)

In these coordinates the constrain (1) is given by λ_{3} = 0. The Hamiltonians of the analysed systems are of the form:

H_{m_{0}} = T + Ṽ_{m_{0}},

where

\begin{aligned}T & = \frac{2}{λ_{1} − λ_{2}} [\frac{(a − λ_{1})(λ_{1} − b)(λ_{1} − c)}{λ_{1}}p_{1}^{2} + \frac{(a − λ_{2})(b − λ_{2})(λ_{2} − c)}{λ_{2}}p_{2}^{2}] \\ p_{j} & = (−1)^{j+1}(λ_{1} − λ_{2})\frac{λ_{j}λ̇_{j}}{4(a − λ_{j})(b − λ_{j})(c − λ_{j})}, j = 1, 2, \end{aligned}

and Ṽ_{m_{0}}(λ_{1}, λ_{2}) we got from V_{m_{0}}(x, y, z) by expressing x^{2}, y^{2}, z^{2} in (2) by (3).

Theorem. The systems with Hamiltonians H_{m_{0}}(λ_{1}, λ_{2}, p_{1}, p_{2}) are integrable by the method of separation of variables in the elliptic coordinate system.

Proof. By the Levi–Civita criterion (see [2, 7]) one has to show that

\begin{matrix}\frac{∂H_{m_{0}}}{∂p_{1}} · \frac{∂H_{m_{0}}}{∂p_{2}} · \frac{∂^{2}H_{m_{0}}}{∂λ_{1}∂λ_{2}} & − \frac{∂H_{m_{0}}}{∂p_{1}} · \frac{∂H_{m_{0}}}{∂λ_{2}} · \frac{∂^{2}H_{m_{0}}}{∂p_{2}∂λ_{1}} \\ −\frac{∂H_{m_{0}}}{∂λ_{1}} · \frac{∂H_{m_{0}}}{∂p_{2}} · \frac{∂^{2}H_{m_{0}}}{∂p_{1}∂λ_{2}} & + \frac{∂H_{m_{0}}}{∂λ_{1}} · \frac{∂H_{m_{0}}}{∂λ_{2}} · \frac{∂^{2}H_{m_{0}}}{∂p_{1}∂p_{2}} = 0 \end{matrix}  (4)

The system with Hamiltonian H_{0} = T is integrable by separation of variables in the elliptical coordinates. So (4) is equivalent to

(λ_{1} − λ_{2})\frac{∂^{2}Ṽ_{m_{0}}}{∂λ_{1}∂λ_{2}} + \frac{∂Ṽ_{m_{0}}}{∂λ_{2}} − \frac{∂Ṽ_{m_{0}}}{∂λ_{1}} = 0.

This can be verifyed directly, using (2) and (3).

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3. Explicit reduction to the Lioville case of separation of variables

We demonstrate explicit reduction of such systems to the Liouville case of separation of variables on the example

V_{−4} = x^{−4}z^{2} + \frac{c(c − a)}{b(b − a)}x^{−4}y^{2}.

Then we give complete integral of the Hamilton-Jacobi equation. Recall that separation of variables in the simpler case of V_{−2} = 1/x^{2} was given by Kozlov in [3].

In elliptic coordinates we have

Ṽ_{−4}(λ_{1}, λ_{2}) = \frac{(a − b)^{2}(c − λ_{1})(c − λ_{2}) − (c − a)^{2}(b − λ_{1})(b − λ_{2})}{(a − λ_{1})^{2}(a − λ_{2})^{2}}.

The following proposition is obtained by simple algebraic transformations.

Proposition. The potential Ṽ_{−4} can be expressed in the form:

Ṽ_{−4}(λ_{1}, λ_{2}) = \frac{1}{λ_{1} − λ_{2}} (φ(λ_{1}) − φ(λ_{2})),

where

φ(λ) = \frac{λ}{a^{2}(a − λ)^{2}} ((a − b)^{2}(c − λ)c − (c − a)^{2}(b − λ)b).

As a corollary we get (see [6]):

Corollary. Complete integral of the Hamilton-Jacobi equation is

\begin{aligned}K & = ∫ \sqrt{\frac{2λ_{1}}{(a − λ_{1})(λ_{1} − b)(λ_{1} − c)} (hλ_{1} − φ(λ_{1}) + α)} dλ_{1} \\ & + ∫ \sqrt{\frac{2λ_{2}}{(a − λ_{2})(b − λ_{2})(λ_{2} − c)} (−hλ_{2} + φ(λ_{2}) − α)} dλ_{2}, \end{aligned}

where α, h are constants representing values of two integrals in involution.

In the case of polynomial potential perturbations (the case m_{0} = 0 from [5]) there is nothing new comparing to the situation considered by Kozlov. The polynomials differ from the Kozlov’s potential V (x, y, z) = k(x^{2} + y^{2} + z^{2}) by the constants, according to the constrained condition (1).

Acknowledgement. The author is grateful to Professor B.A. Dubrovin for helpful suggestions.

References

1. V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Dynamical Systems III, Berlin: Springer, 1988.

2. B.A. Dubrovin, I.M. Krichever and S.P. Novikov, Integrable Systems I in Dynamical Systems IV, Berlin: Spinger, 1988.

3. V.V. Kozlov, Some integrable generalizations of the Jacobi problem on geodesics on the ellipsoid, Prikl. Math. Meh. 59 (1995), 3–9.

4. V.I. Dragovich, On integrable potentials of the billiard system within ellipse, Prikl. Math. Meh. (1996).

5. V.I. Dragovich, On integrable potential perturbations of the Jacobi problem for the geodesics on the ellipsoid, J. Phys. A: Math. Gen. 29 (1996), L317–L321.

6. L.A. Pars, A treatise on Analytical Dynamics, Heinemann, London, 1964.

7. T. Levi-Civita and U. Amaldi, Lezioni di Meccanica Razionale, Nuova edizione, II Parte 2, Bologna, 1952.

Matematički institut (Received 24 10 1996) Kneza Mihaila 35 11001 Beograd, p.p. 367 Yugoslavia