Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

Álvaro Rodríguez Abella; Melvin Leok; Álvaro Rodríguez Abella; Melvin Leok

doi:10.3934/jgm.2023013

Journal of Geometric Mechanics

2023, Volume 15, Issue 1: 319-356. doi: 10.3934/jgm.2023013

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Research article

Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

Álvaro Rodríguez Abella ^{1
,
,},
Melvin Leok ²

1.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera, 13-15, Madrid, 28049, Madrid, Spain
2.
Department of Mathematics, UC San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA

Received: 17 February 2022 Revised: 23 October 2022 Accepted: 09 December 2022 Published: 22 February 2023
37J39, 65P10, 70G65, 70H33

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–Poincaré–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.
- discrete mechanical systems,
- geometric numerical integration,
- Lagrange–Poincaré–Dirac equations,
- reduction by symmetries
Citation: Álvaro Rodríguez Abella, Melvin Leok. Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups[J]. Journal of Geometric Mechanics, 2023, 15(1): 319-356. doi: 10.3934/jgm.2023013

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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–Poincaré–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.

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