Crown relative equilibria for the vortex problem

Abstract

We consider planar central configurations of the κn-vortices problem consisting of κ groups of regular n-gons of equal vorticities, called $(\kappa ,n)$-crown, or equivalently, we study the existence of periodic solutions, called relative equilibrium, for which the vortices rigidly rotate around the center of vorticity, with angular velocity $\lambda \ne 0$. We derive the equations of central configurations for the general $(2,n)$-crown. Next, we give a necessary condition for a $(2,n)$-crown: either the rings are nested (the vertices of the two n-gons are aligned) or they must be rotated by an angle $\pi /n$ (twisted case). After that, we are able to give the exact number of central configurations in function of the ratio of vorticities. More precisely, we show that in the nested case there are two central configurations when the ratio of vorticity is positive, while for a negative ratio of vorticity there exists a unique central configuration for an appropriate radius. For the twisted case, it is observed that the study depends on the number of vortices in each n-gon and the admissible ratio of vorticities must be in an appropriate interval. Our arguments are analytic and differ significantly from the Newtonian case.