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

DOI10.1016/J.JDE.2023.02.066arXiv2211.14820MaRDI QIDQ6418618

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Publication date: 27 November 2022

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 $alpha>0$ and integers $ngeq3$ satisfying $frac1nsu{m}_{j=1}^{n-1}cs{c}^{alpha}fracjpinleq1+fracalpha4$, 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 $nleq6$, the regular $n$-gon is the unique centered co-circular central configuration.

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