On the centered co-circular central configurations for the n-body problem

Abstract

For the power-law potential n-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular central configuration. We get some symmetry results for such central configurations. We show that for positive numbers α>0 and integers n≥3 satisfying 1n∑j=1n−1cscα⁡jπn≤1+α4, the regular n-gon with equal masses is the unique centered co-circular central configuration for the n-body problem with power-law potential Uα. It quickly follows that for the Newtonian n-body problem (in the case α=1) and n≤6, the regular n-gon is the unique centered co-circular central configuration.