Almost-Poisson brackets for nonholonomic systems with gyroscopic terms and Hamiltonisation

TL;DR

This paper develops a geometric framework for nonholonomic systems with gyroscopic terms, extending almost-Poisson brackets and Hamiltonisation methods to include affine Legendre transforms, relevant for control and rigid body dynamics.

Contribution

It introduces a novel geometric formalism for nonholonomic systems with gyroscopic terms, accounting for affine Legendre transforms and brackets, advancing Hamiltonisation techniques.

Findings

Constructed affine-dependent brackets for systems with gyroscopic terms

Provided a geometric framework for Hamiltonisation of Suslov and Chaplygin sphere problems

Extended almost-Poisson structures to non-mechanical Lagrangians

Abstract

We extend known constructions of almost-Poisson brackets and their gauge transformations to nonholonomic systems whose Lagrangian is not mechanical but possesses a gyroscopic term linear in the velocities. The new feature introduced by such a term is that the Legendre transformation is an affine, instead of linear, bundle isomorphism between the tangent and cotangent bundles of the configuration space and some care is needed in the development of the geometric formalism. At the end of the day, the affine nature of the Legendre transform is reflected in the affine dependence of the brackets that we construct on the momentum variables. Our study is motivated by a wide class of nonholonomic systems involving rigid bodies with internal rotors which are of interest in control. Our construction provides a natural geometric framework for the (known) Hamiltonisations of the Suslov and Chaplygin sphere problems with a gyrostat.