Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

TL;DR

This paper develops a comprehensive theory for discrete Dirac reduction of Lagrangian systems with abelian symmetry, extending linear to nonlinear cases using retraction charts, and demonstrates applications to physical systems.

Contribution

It introduces a unified approach to discrete Dirac reduction for both linear and nonlinear systems with abelian symmetry, utilizing discrete connections and principal bundles.

Findings

Derived discrete Lagrange-Poincaré-Dirac equations

Unified reduction of Dirac structures and Lagrange-Pontryagin principles

Applied method to charged particle and double spherical pendulum

Abstract

This paper develops the theory of discrete Dirac reduction of discrete Lagrange-Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange-Pontryagin principle, and show that they both lead to the same discrete Lagrange-Poincar\'e-Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last, we demonstrate the method obtained by applying it to a charged particle in a magnetic field, and to the double spherical pendulum.