The stability of Riemann ellipsoids in the Lyapunov formulation

Abstract

The problem of the stability of the Riemann ellipsoids of a rotating uniform self-gravitating ideal liquid is considered within the framework of the Lyapunov definition of the stability of the form of equilibrium [1]. The regions such that almost all the ellipsoids belonging to it are unstable forms of equilibrium, specified in explicit analytical form, are determined in parameter spaces of the first and second families of Riemann ellipsoids. The proof is based on the general fact (which is formulated and justified separately) that, when an unstable equilibrium position of an autonomous system is stable with respect to a certain function, the trajectory of this system, which belongs to a certain manifold, is obtained, and also on a consequence of this fact, which has a constructive form. The stability of the form of ellipsoidal figures of equilibrium, with the exception of special cases of Maclaurin and Jacobi ellipsoids, the stability of the form of which was investigated by Lyapunov himself, has not been investigated previously in the literature.