N-particle dynamics of the Euler equations for planar diffeomorphisms

The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this article, we consider the dynamics of N point particles or “blobs” moving under the action of the Euler equations associated with the group of diffeomorphisms of the plane in a variety of different metrics. This dynamical system is already in widespread use in the field of image registration, where the point particles correspond to image landmarks, but its dynamical behavior has not previously been studied. The 2-body problem is always integrable, and we analyze its phase portrait under different metrics. In particular, we show that 2-body capturing orbits (in which the distances between the particles tend to 0 as t  → ∞) can occur when the kernel is sufficiently smooth and the relative initial velocity of the particles is sufficiently large. We compute the dynamics of these “dipoles” with respect to other test particles, and supplement the calculations with simulations for larger N that illustrate the different regimes.