Discrete Lagrange Problems with Constraints Valued in a Lie Group

TL;DR

This paper develops a discrete variational framework for Lagrange problems with Lie group constraints, establishing a multiplier rule, symmetry theory, and applications to discrete Euler-Poincaré reduction and harmonic maps.

Contribution

It introduces a novel discrete Lagrange multiplier approach with Lie group constraints, including a Cartan form and Noether theory, applied to discrete field theories.

Findings

Critical sections satisfy an unconstrained variational problem.

A discrete Noether theorem and multisymplectic form formula are established.

Application to harmonic maps in SO(n) demonstrates the theory's effectiveness.

Abstract

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincar\'e reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group $SO(n)$.