Spherical and Planar Ball Bearings — a Study of Integrable Cases

Abstract

We consider the nonholonomic systems of n homogeneous balls B1,...,Bn with the same radius R that are rolling without slipping about a fixed sphere S0 with center O and radius R.In addition, it is assumed that a dynamically nonsymmetric sphere S with the center that coincides with the center O of the fixed sphere S0 rolls withoutslipping in contact with the moving balls B1,...,Bn. The problem is considered in four different configurations, three of which are new.We derive the equations of motion and find an invariant measure for these systems.As the main result, for n=1 we find two cases that are integrable by quadratures according to the Euler – Jacobi theorem.The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems.Further, we explicitly integratethe planar problem consisting of n homogeneous balls of the same radius, but with differentmasses, which roll without slippingover a fixed plane Σ0 with a plane Σ that moves without slipping over these balls.