Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups

We formulate Euler-Poincaré equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EP A ut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing Δ-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group Aut _vol(P) of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.