  We study invariant manifolds of measure-valued solutions of the partial
differential equation for geodesic flow of a pressureless fluid. These
solutions describe interaction dynamics on lower-dimensional support sets; for
example, curves, or filaments, of momentum in the plane. The 2+1 solutions we
study are planar generalizations of the 1+1 peakon solutions of Camassa & Holm
[1993] for shallow water solitons. As an example, we study the canonical
Hamiltonian interaction dynamics of $N$ rotating concentric circles of peakons,
whose solution manifold is $2N-$dimensional. Thus, the problem is reduced from
infinite dimensions to a finite-dimensional, canonical, invariant manifold. The
existence of this reduced solution manifold and many of its properties may be
understood, by noticing that it is also the momentum map for the action of
diffeomorphisms on the space of curves in the plane. We show both analytical
and numerical results.
