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The EPDiff equation (or dispersionless Camassa-Holm equation in 1D) is a wellknown example of geodesic motion on the Diff group of smooth invertible maps(diffeomorphisms). Its recent two-component extension governs geodesic motionon the semidirect product ${\rm Diff}\circledS{\cal F}$, where $\mathcal{F}$denotes the space of scalar functions. This paper generalizes the secondconstruction to consider geodesic motion on ${\rm Diff} \circledS\mathfrak{g}$,where $\mathfrak{g}$ denotes the space of scalar functions that take values ona certain Lie algebra (for example,$\mathfrak{g}=\mathcal{F}\otimes\mathfrak{so}(3)$). Measure-valued delta-likesolutions are shown to be momentum maps possessing a dual pair structure,thereby extending previous results for the EPDiff equation. The collectiveHamiltonians are shown to fit into the Kaluza-Klein theory of particles in aYang-Mills field and these formulations are shown to apply also at thecontinuum PDE level. In the continuum description, the Kaluza-Klein approachproduces the Kelvin circulation theorem.

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