Reduction in principal fiber bundles: covariant Euler-Poincare equations

Abstract

Let $\pi :P\to M^n$ be a principal G-bundle, and let $\mathcal{L}:J^{1}P\to {\Lambda }^n(M)$ be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on $\mathcal{C}(P)$, the bundle of connections on P.