Spherical and Planar Ball Bearings -- a Study of Integrable Cases

TL;DR

This paper studies nonholonomic systems of rolling balls around a fixed sphere, deriving equations of motion, proving invariant measures, and identifying integrable cases for one ball, extending classical Chaplygin ball problems.

Contribution

It introduces new integrable nonholonomic models involving multiple rolling balls and extends classical problems to more complex configurations.

Findings

Derived equations of motion for the systems.

Proved the existence of invariant measures.

Identified two integrable cases for a single ball.

Abstract

We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping in contact to the moving balls $\mathbf B_1,\dots,\mathbf B_n$. The problem is considered in four different configurations. We derive the equations of motion and prove that these systems possess an invariant measure. As the main result, for $n=1$ we found two cases that are integrable in 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, that roll without slipping over a fixed plane $\Sigma_0$ with a plane $\Sigma$ that moves without slipping over these balls.