On Motion of Rigid Bodies on a Spherical Surface

The problem on description of dynamics of mechanical systems performing motions in spaces of a constant curvature is known in mechanics. Its investigation can be followed since the publication of Zhukovsky [1], devoted to the problem on motion of a plate on a surface of a pseudo-sphere which was proposed even by Lobachevsky. In recent publications [2, 3] the studies of Zhukovsky were developed. In particular, in the problem on motion of a massive point at a sphere and at a pseudo-sphere in a field of an attracting center there were found the analogs of Kepler's laws, there was studied Bertrand's problem concerning a description of all central force fields, for which all trajectories are closed. There was also integrated a problem on two attracting centers. These studies were continued in [4], where the questions on integrability of the problem on two Newtonian centers in three-dimensional spaces of negative and positive constant curvature as well as on existence of steady motions of two bodies under mutual attraction in these space were considered. In this paper the more general problem on motion of axisymmetric rigid bodies on the surface of a three-dimensional sphere is considered. Under appropriate assumptions these bodies can be treated as "spherical" planets. The comparison of dynamics of axisymmetric rotating planet with dynamics of analoguous system in a flat space is carried out.