Euler-Poincare reduction for discrete field theories

TL;DR

This paper develops a discrete Euler-Poincare reduction framework for field theories, connecting discrete differential geometry with variational principles, and extends existing schemes with applications to harmonic mappings.

Contribution

It introduces a novel Euler-Poincare reduction method for discrete field theories and demonstrates their equivalence with an extended Moser-Veselov scheme.

Findings

Discrete Euler-Poincare equations are formulated.

The equivalence between the new equations and the extended Moser-Veselov scheme is established.

Application to discrete harmonic mappings is discussed.

Abstract

In this note, we develop a theory of Euler-Poincare reduction for discrete Lagrangian field theories. We introduce the concept of Euler-Poincare equations for discrete field theories, as well as a natural extension of the Moser-Veselov scheme, and show that both are equivalent. The resulting discrete field equations are interpreted in terms of discrete differential geometry. An application to the theory of discrete harmonic mappings is also briefly discussed.