  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 paper, 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. 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 $t \to \infty$) 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.
