Alexandr Burov

The problem of motion of a heavy particle on a sphere uniformly rotating about a fixed axis is considered in the case of dry friction. It is assumed that the angle of inclination of the rotation axis is constant. The existence of equilibria in an absolute coordinate system and their linear stability are discussed. The equilibria in a relative coordinate system rotating with the sphere are also studied. These equilibria are generally nonisolated. The dependence of the equilibrium sets of this kind on the system parameters is also considered.

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.

It is well-known in the Theoretical Mechanics how using the Routh procedure one can reduce the order of the Lagrange equations describing the motion if the cyclic coordinates expressing the symmetry properties of the mechanical system are known [1,2]. However, if the equations of motion are written in redundant variables then the procedure of reduction is not always obvious The idea of the reduction for such systems can be traced back to Lyapunov (see [3], p. 353–355), who proposed to consider a motion with respect to the rotating, in the general case nonuniformly, specially chosen frame in the problem on gures of equilibria of the rotating fluid. The development of the studies of Lyapunov was given in [4].
As it is known the general method of such reduction for equations of Poincaré–Chetayev was proposed by Chetayev [5,6], see also [7]. However the realisation of the Chetayevs theorem on reduction is not always simple for real systems. In this paper the analogue of the Routh procedure is considered for the problem on motion of mechanical system consisting of the rigid body with the fixed point. The origine of the concept of the reduced (amended) potential is shown. The problem on motion of the anedeformable body is considered in details.