Uniqueness results for co-circular central configurations for power-law potentials

For a class of potential functions including those used for the planar n -body and n -vortex problems, we investigate co-circular central configurations whose center of mass coincides with the center of the circle containing the bodies. Useful equations are derived that completely describe the problem. Using a topological approach, it is shown that for any choice of positive masses (or circulations), if such a central configuration exists, then it is unique. It quickly follows that if the masses are all equal, then the only solution is the regular n -gon. For the planar n -vortex problem and any choice of the vorticities, we show that the only possible co-circular central configuration with center of vorticity at the center of the circle is the regular n -gon with equal vorticities.