On novel Hamiltonian description of the nonholonomic Suslov problem

TL;DR

This paper introduces new Poisson structures for the nonholonomic Suslov problem, providing a Hamiltonian framework with invariant Poisson bivectors and Casimir functions, enhancing understanding of its geometric properties.

Contribution

The authors develop novel Poisson bivectors with invariant properties and Casimir functions, offering a new Hamiltonian perspective on the Suslov problem and its gyrostat variant.

Findings

Two rank four invariant Poisson bivectors with cubic brackets

Rank two Poisson bivectors with two Casimir functions for the gyrostat case

New geometric structures invariant under the Suslov flow

Abstract

We present some new Poisson bivectors that are invariants by the flow of the nonholonomic Suslov problem. Two rank four invariant Poisson bivectors have globally defined Casimir functions and, therefore, define cubic Poisson brackets on the five dimensional state space with standard symplectic leaves. For the Suslov gyrostat in the potential field we found rank two Poisson bivectors having only two globally defined Casimir functions and, therefore, we say about formal Hamiltonian description in these cases.