Spherical and planar ball bearings -- nonholonomic systems with invariant measures

TL;DR

The paper constructs and analyzes nonholonomic systems of rolling balls, proving they have invariant measures and integrability properties, both in spherical and planar limits, contributing to the understanding of such complex mechanical systems.

Contribution

It introduces new nonholonomic models of rolling balls with invariant measures and demonstrates their integrability in both spherical and planar configurations.

Findings

Systems possess invariant measures

Planar limit systems are integrable in quadratures

Models extend understanding of nonholonomic rolling systems

Abstract

We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $\Sigma_0$, and a moving plane $\Sigma$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler-Jacobi theorem.