Poisson structures for reduced non-holonomic systems

TL;DR

This paper investigates the algebraic structure of Poisson brackets in non-holonomic systems, revealing their rank and generalizing their form, which helps understand Hamiltonian structures in rolling body problems.

Contribution

It demonstrates that certain Poisson structures are of rank two and generalizes their form, extending their applicability to various rolling body systems.

Findings

Poisson structures are of rank two, eliminating the need for rescaling.

Poisson structures are determined by first integrals and differential equations.

The theory applies to multiple rolling body problems, including spheres and cylinders.

Abstract

Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank two and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing two integrals of motion. We generalize the form of that Poisson structures and extend their domain of definition. We apply the theory to the rolling disk, the Routh's sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder.