Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

Abstract

We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an n-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on n parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of symmetry, noncommutative integrability for any ratio of the radii is established.