Reduction in principal fiber bundles: covariant Euler-Poincare equations

Let $\pi:P\to M^n$ be a principal G-bundle, and let ${\mathcal{L}}: J^1P \to\Lambda^n(M)$ 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.