Euler-Poincaré reduction for discrete field theories

Abstract

In this note, we develop a theory of Euler-Poincaré reduction for discrete Lagrangian field theories. We introduce the concept of Euler-Poincaré 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.