An L–A pair for the Hess–Apel'rot system and a new integrable case for the Euler–Poisson equations on so(4) × so(4)

Abstract

We present an L{A pair for the Hess{Apel' rot case of a heavy rigid three-dimensional body. Using it, we give an algebro-geometric integration procedure. Generalizing this L{A pair, we obtain a new completely integrable case of the Euler{Poisson equations in dimension four. Explicit formulae for integrals that are in involution are given. This system is a counterexample to one of Ratiu' s theorems. A corrected version of this classi¯cation theorem is proved.

The Lagrange bitop on so (4) × so (4) and geometry of the Prym varieties

“…The general orbits of the coadjoint action are 8 dimensional. According to [25], the Casimir functions are coefficients So we have Proposition 2 [11]. For |χ 12 | = |χ 34 |, the system (3, 4) is completely integrable in the Liouville sense.…”

“…The generalized Liouville tori are four dimensional. Since two of the integrals of the motion of the Lagrange bitop are linear (see (11)), according to the well known fact of Classical Mechanics ( [31,3]) those generalized tori have twodimensional affine part. The two-dimensional compact part of such a torus corresponds to the real part of the two-dimensional Prymian Π.…”

On the cases of Kirchhoff and Chaplygin of the Kirchhoff equations

“…and α, β are constants that satisfy α 2 + β 2 = 1. The expressions (4), (5) are the first terms in the Laurent series for a solution…”

“…The unperturbed system coincides with the unperturbed system of equations (3). Hence, it has a particular solution (4), (5). For the terms of order ǫ, one gets the system which homogeneous part is (7).…”

Asymptotic behaviour of singular points of solutions of the problem of heavyn-dimensional body motion in the Lagrange case

“…Moreover, no general method for finding an L − A pair is known, and when it can be found (see [7]), not necessarily all the solutions of the problem are obtained, as we see in the example of the Hess problem discussed by Dragović and Gajić in [8].…”