  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 ${\rm Diff} \circledS\mathfrak{g}$,
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.
