On polygonal relative equilibria in the N-vortex problem

Helmholtz’s equations provide the motion of a system of N vortices which describes a planar incompressible fluid. A relative equilibrium is a particular solution of these equations for which the distances between the particles are invariant during the motion. In this article, we are interested in relative equilibria formed of concentric regular polygons of vortices. We show that in the case of one regular polygon with more than three vertices, a relative equilibrium requires equal vorticities. This result is the analogous of Perko-Walter-Elmabsout’s in celestial mechanics. We also compute the number of relative equilibria with two concentric regular n-gons and the same vorticity on each n-gon, and we study the associated co-rotating points. This result completes previous studies for two regular n-gons when all the vortices have the same vorticity and when the total vorticity vanishes.