One invariant measure and different Poisson brackets for two nonholonomic systems

TL;DR

This paper explores two nonholonomic systems, analyzing their symplectic forms and Poisson structures, revealing differences from classical constructions and providing new insights into their geometric properties.

Contribution

It introduces novel Poisson bivectors for nonholonomic systems using L-tensors with torsion, contrasting with traditional geometric frameworks.

Findings

Different symplectic forms for the systems studied

Poisson bivectors determined by L-tensors with torsion

Contrasts with Eisenhart-Benenti and Turiel constructions

Abstract

We discuss the nonholonomic Chaplygin and the Borisov-Mamaev-Fedorov systems, for which symplectic forms are different deformations of the square root from the corresponding invariant volume form. In both cases second Poisson bivectors are determined by $L$-tensors with non-zero torsion on the configurational space, in contrast with the well known Eisenhart-Benenti and Turiel constructions.