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

Abstract

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Ⓢ , where denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on DiffⓈ , where denotes the space of scalar functions that take values on a certain Lie algebra (e.g. = ⊗ (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.