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

TL;DR

This paper introduces a unified approach to time reparametrization methods for nonholonomic systems, demonstrating integrability of a rolling ball over a sphere under specific conditions and symmetries.

Contribution

It extends the Chaplygin multiplier method and Maupertuis principle to a unified framework, proving integrability for a class of nonholonomic rolling problems.

Findings

Complete integrability when the ball's radius is twice the sphere's radius.

Noncommutative integrability for any ratio of radii under certain symmetries.

Unified framework for time reparametrization in nonholonomic mechanics.

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 $SO(l)\times SO(n-l)$ symmetry, noncommutative integrability for any ratio of the radii is established.