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

The EPDiff equation (or dispersionless Camassa-Holm equation in 1D) 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 ${rm Diff}circledS{cal F}$, where $mathcal{F}$ denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on ${rm Diff} circledSmathfrak{g}$, where $mathfrak{g}$ denotes the space of scalar functions that take values on a certain Lie algebra (for example, $mathfrak{g}=mathcal{F}otimesmathfrak{so}(3)$). 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 PDE level. In the continuum description, the Kaluza-Klein approach produces the Kelvin circulation theorem.