Invariant sets in the Clebsch-Tisserand problem: Existence and stability

Questions of the existence and stability of invariant sets in the problem of the motion of a rigid body with a fixed point in an axi-symmetric force field with a quadratic potential (with respect to the direction cosines of the axis of symmetry of the field) are discussed. This problem is isomorphic with the problem of the motion of a free rigid body bounded by a simply connected surface in an ideal homogeneous incompressible fluid which performs irrotational motion and is at rest at infinity. [Kirchhoff F G.R. Uber die Bewegung eines Rotationskorpers in eine Flussigkeit, J Reine und angew Math, B. 71, 1870, S.237-S.262; Clebsch A. Uber die Bewegung eines Koipers in eine Flussigkeit, Math Anmnalen, Bd. 3, 1870, S.238-S.262] In particular, in the second case of the Clebsch integrability of a problem of the motion of a body in a fluid, this problem is isomorphic with the Tisserand problem on the motion of a body with a fixed point in the case of a quadratic potential of special form,[Tisserand M.F. Sur le mouvement des planetes autour du Soleil d'aprees la lol electrodynamique de Weber, C.R. Acad. Sci. Paris, 1872. V. 75, P. 760-763] which corresponds to the satellite approximation of the potential of the forces of Newtonian attraction. (c) 2007 Elsevier Ltd. All rights reserved.