Rolling of a Homogeneous Ball on a Moving Cylinder

This paper addresses the problem of a homogeneous ball rolling on the inner surface of a circular cylinder in a field of gravity parallel to its axis. It is assumed that the ball rolls without slipping on the surface of the cylinder, and that the cylinder executes plane-parallel motions in a circle perpendicular to its symmetry axis. The integrability of the problem by quadratures is proved. It is shown that in this problem the trajectories of the ball are quasiperiodic in the general case, and that an unbounded elevation of the ball is impossible. However, in contrast to a fixed (or rotating) cylinder, there exist resonances at which the ball moves on average downward with constant acceleration.