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| Due to symmetry, the equal mass regular $n$-gon is a central configuration for the $n$-body problem, the $n$-vortex problem, and similarly constructed systems. In an article on perverse choreographies, Alain Chenciner asked if the regular $n$-gon was the only central configuration lying on a circle, with the additional property that the center of the circle coincides with the center of mass. Using a topological approach, we show 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. |
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