The local description of discrete Mechanics

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-Poincaré equations, discrete Lagrange-Poincaré 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.