Integrability of Nonholonomically Coupled Oscillators

TL;DR

This paper investigates a family of nonholonomic harmonic oscillators, demonstrating their integrability and reversibility, which explains their favorable numerical properties and relevance as models for transmission gearboxes.

Contribution

The paper proves that all systems in this family are integrable and reversible, providing new insights into their structure and numerical behavior.

Findings

All systems are integrable and reversible.

Numerical discretizations preserve structure.

Models relate to transmission gearboxes.

Abstract

We study a family of nonholonomic mechanical systems. These systems consist of harmonic oscillators coupled through nonholonomic constraints. In particular, the family includes the so called contact oscillator, which has been used as a test problem for numerical methods for nonholonomic mechanics. Furthermore, the systems under study constitute simple models for continuously variable transmission gearboxes. The main result is that each system in the family is integrable reversible with respect to the canonical reversibility map on the cotangent bundle. This result may explain previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.