The local description of discrete Mechanics

TL;DR

This paper develops local geometric expressions for discrete Mechanics within the framework of Lie groupoids, facilitating the explicit construction of geometric integrators for various mechanical systems.

Contribution

It introduces local descriptions of discrete Mechanics on Lie groupoids, extending previous global studies and aiding in the development of geometric integrators.

Findings

Local expressions for discrete Mechanics are derived.

These local descriptions help construct geometric integrators.

Applications include discrete Euler-Lagrange and Euler-Poincaré equations.

Abstract

In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by Alan Weinstein in [19]. To do this, we will need some results on the geometry of Lie groupoids, as, for instance, the construction of symmetric neighborhoods or the existence of local bisections. These local descriptions will be particular useful for the explicit construction of geometric integrators for mechanical systems (reduced or not), in particular, discrete Euler-Lagrange equations, discrete Euler-Poincar\'e equations, discrete Lagrange-Poincar\'e equations... The results contained in this paper can be considered as a local version of the study that we have started in [13], on the geometry of discrete Mechanics on Lie groupoids.