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

TL;DR

This paper extends the geometric framework of the EPDiff equation to more general semidirect product Lie groups, revealing measure-valued solutions as momentum maps and connecting the theory to Kaluza-Klein and Yang-Mills physics.

Contribution

It generalizes the geodesic flow on semidirect product groups to include Lie algebra-valued functions, and shows measure-valued solutions as momentum maps with a dual pair structure.

Findings

Measure-valued solutions are momentum maps with dual pair structure.

The Hamiltonian framework aligns with Kaluza-Klein theory and Yang-Mills fields.

The continuum PDE formulation includes a Kelvin circulation theorem.

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 ${\rm Diff}\circledS{\cal F}$, 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.