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

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 $\alpha>0$ and integers $n\geq3$ satisfying $\frac{1}{n}\sum_{j=1}^{n-1}\csc^{\alpha}\frac{j\pi}{n}\leq1+\frac{\alpha}{4}$, the regular $n$-gon with equal masses is the unique centered co-circular central configuration for the $n$-body problem with power-law potential $U_{\alpha}$. It quickly follows that for the Newtonian $n$-body problem (in the case $\alpha=1$) and $n\leq6$, the regular $n$-gon is the unique centered co-circular central configuration.