A note on the energy of relative equilibria of point vortices

The problem of determining relative equilibria of identical point vortices is longstanding. The simplest such equilibria, vortices arranged at the vertices of a regular polygon with or without one at the center, go back to the work by Kelvin and Thomson late in the 19th century, and are wound up with the now defunct theory of vortex atoms. Many years later, Havelock found relative equilibria consisting of two nested, regular polygons. One may place a vortex at the center of such configurations !modulo adjustment of the radii" and produce further relative equilibria. More recently Aref and van Buren found relative equilibria consisting of three nested regular polygons. An important quantity to be determined for all such configurations is the kinetic energy of the induced flow, which is given by the point vortex Hamiltonian. For identical point vortices, this is, in essence, a purely geometric quantity, viz. !the logarithm of" the product of all intravortex distances in the configuration. For N vortices, then, we wish to calculate the product of the N!N−1" /2 vortex distances. The main result of this paper is a formula, given as Eq. !14" below, that allows the analytical determination of this product for the analytically known relative equilibria. Let us establish a bit of notation. Let the vortices be given as N points in the complex plane %z! &!=1, . . . ,N'. In order to form a relative equilibrium configuration for N identical point vortices, the z! must solve the system of equations,