Relative equilibria of vortices in two dimensions

An old problem of the evolution of finitely many interacting point vortices in the plane is shown to be amenable to investigation by critical point theory in a way that is identical to the study of the planarn -body problem of celestial mechanics. For any choice of positive circulations of the vortices it is shown by critical point theory applied to Kirchhoff's function that there are many relative equilibria configurations. Each of these configurations gives rise to a stationary configuration of the vortices in a suitably chosen rotating coordinate system. A sharp lower bound on the number of stationary vortex configurations for the problem of point vortices interacting in the plane is given. The problem of point vortices in a circular disk is defined and it is shown that these estimates hold for stationary configurations of small size.