Nonholonomic systems with symmetry allowing a conformally symplectic reduction

TL;DR

This paper investigates nonholonomic systems with symmetry, demonstrating that under certain conditions, the reduced system becomes conformally symplectic, providing new insights into their geometric structure.

Contribution

It shows that in simple nonholonomic systems with symmetry, the compressed system can be conformally symplectic, even though the original constrained system lacks a Jacobi structure.

Findings

Compressed systems are conformally symplectic under certain conditions

Original constrained systems do not admit a Jacobi structure

Symmetry plays a key role in the reduction process

Abstract

Non-holonomic mechanical systems can be described by a degenerate almost-Poisson structure (dropping the Jacobi identity) in the constrained space. If enough symmetries transversal to the constraints are present, the system reduces to a nondegenerate almost-Poisson structure on a ``compressed'' space. Here we show, in the simplest non-holonomic systems, that in favorable circumnstances the compressed system is conformally symplectic, although the ``non-compressed'' constrained system never admits a Jacobi structure (in the sense of Marle et al.).