Ball bearings as generalizations of the Chaplygin ball problem

Abstract

The motion of n homogeneous balls with the same radius that are moving without slipping over a fixed sphere is considered. In addition, it is assumed that a dynamically nonsymmetric sphere rolls without slipping in contact to the moving balls. The centers of the moving and the fixed spheres coincide. Four different configurations are considered. We prove that the equations of motion admit invariant measure for these systems. For n = 1 we found two integrable cases. The obtained integrable nonholonomic models represent natural extensions of the well known Chaplygin ball integrable problem.