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

Abstract

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 $\mathrm{Diff}Ⓢ\mathcal{F}$, where $\mathcal{F}$ denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on $\mathrm{Diff}Ⓢ\mathfrak{g}$, where $\mathfrak{g}$ denotes the space of scalar functions that take values on a certain Lie algebra (for example, $\mathfrak{g}=\mathcal{F}\otimes \mathfrak{so}(3)$). 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.

22 pages, 2 figures. Submitted to Proc. R. Soc. A

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