Simple non-Hamiltonian systems with an invariant measure

TL;DR

This paper introduces a straightforward method to construct non-Hamiltonian dynamical systems that maintain an invariant measure, linking them to known nonholonomic systems like the Chaplygin ball and Veselova system.

Contribution

It presents a novel construction approach for non-Hamiltonian systems with invariant measures, extending the understanding of their relation to Hamiltonian deformations and nonholonomic systems.

Findings

Constructed non-Hamiltonian systems with invariant measures.

Identified connections to known nonholonomic systems.

Illustrated results with Chaplygin ball and Veselova system.

Abstract

We propose a simple construction of the non-Hamiltonian dynamical systems possessing an invariant measure. These non-Hamiltonian systems are deformations of the Hamiltonian systems associated with trivial deformations of the canonical Poisson brackets and time transformation. Sometimes these non-Hamiltonian systems can be identified with known nonholonomic systems and, therefore, the results are illustrated through the Chaplygin ball and the Veselova system.