Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or the dispersionless Camassa-Holm equation in one dimension) is a well-known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product Diff(S)F, where F denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on Diff(S)g, where g denotes the space of scalar functions that take values on a certain Lie algebra (e. g. g = F circle times so(3)). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza-Klein theory of particles in a Yang-Mills field and these formulations are shown to apply also at the continuum partial differential equation level. In the continuum description, the Kaluza-Klein approach produces the Kelvin circulation theorem.