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

2022/11/27 by Zhi-Qiang Wang, Wang, Zhiqiang
Engineering · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Nuclear physics research studies #Spacecraft Dynamics and Control #Stellar, planetary, and galactic studies

paper · pdf · doi:10.48550/arxiv.2211.14820

openalex publication_date 2022/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (1)/(n)∑_j=1^n-1\csc^α(jπ)/(n)≤1+\fracα4, 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_α. 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.

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