Lagrangian reduction of nonholonomic discrete mechanical systems by stages

TL;DR

This paper introduces a new categorical framework for discrete Lagrangian systems, extending reduction methods for nonholonomic systems with symmetries, and proves the equivalence of one-stage and two-stage reduction processes.

Contribution

It develops the category $LDP_d$ for discrete systems, defines symmetry and reduction within it, and shows two-stage reduction is equivalent to one-stage reduction under certain conditions.

Findings

Defined the category $LDP_d$ for discrete systems.

Extended reduction methods to include two-stage processes.

Proved equivalence of one-stage and two-stage reductions.

Abstract

In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincar\'e systems, and study some of its elementary properties. Examples of objects of $LDP_d$ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincar\'e systems. We also introduce a notion of symmetry group for objects of $LDP_d$ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange--Poincar\'e systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in $LDP_d$ to the system obtained by a one-stage reduction by the full symmetry group.