Spherical and Planar Ball Bearings — a Study of Integrable Cases

We consider the nonholonomic systems of n homogeneous balls 𝐁_1,…,𝐁_n with the same radius r that are rolling without slipping about a fixed sphere 𝐒_0 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 𝐒_0 rolls without slipping in contact with the moving balls 𝐁_1,…,𝐁_n . 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 integrate the planar problem consisting of n homogeneous balls of the same radius, but with different masses, which roll without slipping over a fixed plane Σ_0 with a plane Σ that moves without slipping over these balls.