Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics

TL;DR

This paper introduces a high-order structure-preserving discretization method for Hamiltonian systems that are reparametrized in time, with applications to nonholonomic mechanics, ensuring measure preservation and accurate flow approximation.

Contribution

It presents the first measure-preserving discretization for measure-preserving nonholonomic systems, extending Hamiltonian discretization techniques to a broader class of systems.

Findings

Discretization preserves smooth measure on phase space to arbitrary order.

The method accurately interpolates the flow of the continuous system.

Application to nonholonomic systems allows Hamiltonization and measure preservation.

Abstract

We propose a discretization of vector fields that are Hamiltonian up to multiplication by a positive function on the phase space that may be interpreted as a time reparametrization. We prove that our method is structure preserving in the sense that the discrete flow is interpolated to arbitrary order by the flow of a continuous system possessing the same structure. In particular, our discretization preserves a smooth measure on the phase space to arbitrary order. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, these results provide the first occurrence in the literature of a measure preserving discretization of measure preserving nonholonomic systems.