Integrability of nonholonomically coupled oscillators

Klas Modin; Olivier Verdier

doi:10.3934/dcds.2014.34.1121

Article Contents

2014, Volume 34, Issue 3: 1121-1130. Doi: 10.3934/dcds.2014.34.1121

This issue Previous Article On an asymptotic method for computing the modified energy for symplectic methods Next Article Regarding the absolute stability of Størmer-Cowell methods

Integrability of nonholonomically coupled oscillators

Klas Modin^1, and
Olivier Verdier^2,

1.
Department of Mathematical Sciences, Chalmers University of Technology
2.
Department of Mathematics, University of Bergen

Received: December 2012

Revised: February 2013

Published: March 2014

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Abstract

We study a family of nonholonomic mechanical systems. These systems consist of harmonic oscillators coupled through nonholonomic constraints. The family includes the contact oscillator, which has been used as a test problem for numerical methods for nonholonomic mechanics. 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. By using reversible Kolmogorov--Arnold--Moser theory, we then establish preservation of invariant tori for reversible perturbations. This result explains previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.

Keywords:

Mathematics Subject Classification: Primary: 70H08, 70F25, 37J60; Secondary: 65P99, 37M99.

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