Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces

TL;DR

This paper explores conditions under which a modified form of energy remains conserved in certain nonholonomic systems with affine constraints, and applies these ideas to analyze the integrability of a rolling sphere on a rotating surface.

Contribution

It introduces a conserved energy-like function for nonholonomic systems with affine constraints and demonstrates its role in analyzing the integrability of a rolling sphere on rotating surfaces.

Findings

A modified energy function can be conserved under specific conditions.

Symmetry plays a key role in the conservation mechanism.

The sphere's motion can be quasi-periodic on tori of up to three dimensions.

Abstract

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.