Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin hamiltonization

Abstract

In this note we consider the nonholonomic problem of rolling without slipping and twisting of an n-dimensional balanced ball over a fixed sphere. This is a SO(n)-Chaplygin system with an invariant measure that reduces to the cotangent bundle T* Sn-1. For the rigid body inertia operator Iω = Iω + ωI, I = diag(I1, ..., In) with a symmetry I1 = I2 = ··· = Ir ≠ Ir+1 = Ir+2 = ··· = In, we prove that the reduced system is integrable, general trajectories are quasi-periodic, while for r ≠ 1, n - 1 the Chaplygin reducing multiplier method does not apply.