On periodic solutions of the euler — poisson equations: PMM vol. 43, no. 2, 1979, pp. 262–266

Abstract

A novel family is shown of the periodic solutions of the equations of motion of a heavy rigid body about a fixed point, for the case when the mass distribution within the body is almost the same as the distribution in the case of the Lagrange integrability. The periodic solutions obtained for the unperturbed problem correspond to the case when one of the frequencies of the regular Lagrange precession becomes equal to zero, i.e. they correspond to the steady rotations about the axes situated in the principal plane of inertia of the body about the fixed point. These steady rotations correspond to a bifurcation, and the regular precessions branch out from them [1]. For such motions the variational equations of the problem stated above have two zero roots with a single group of solutions. The periodic solutions near to the steady rotations of an arbitrary solid, were studied in [2]. These solutions, as well as the solutions obtained in [3], were based on the Liapunov theorem, i.e. on the assumption that the frequencies of the corresponding system of equations in variations were incommensurable. The author of [4] used the Poincaré method to show the existence of periodic solutions generated by the steady rotations of the Euler case.