Dynamics of the Chaplygin ball on a rotating plane

TL;DR

This paper studies the complex chaotic dynamics of a Chaplygin ball rolling on a rotating plane, revealing the existence of invariant measures and chaotic behavior through a reduced three-dimensional Poincare map.

Contribution

It introduces a new analysis of the Chaplygin ball problem on a rotating plane, highlighting the chaotic nature of the system and its invariant measures.

Findings

The system admits area integrals and an invariant measure.

The reduced Poincare map preserves phase volume.

The system exhibits chaotic dynamics in the general case.

Abstract

This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincare map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system's dynamics is chaotic.