Two integrable cases of a ball rolling over a sphere in Rn

Abstract

We consider the nonholonomic problem of rolling without slipping and twisting of a balanced ball over a fixed sphere in Rn. By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover, this is a SO(n)-Chaplygin system that reduces to the cotangent bundle T∗Sn-1. We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system. In the second case, we consider the rigid body inertia operator Iω = Iω + ωI, I = diag(I1, . , In) with a symmetry I1 = I2 = . = Ir ≠= Ir+1 = Ir+2 = . = In. It is shown that general trajectories are quasi-periodic, while for r ≠= 1, n-1 the Chaplygin reducing multiplier method does not apply.