Super central configurations of the three-body problem under the inverse integer power law

Abstract

In this paper, we consider the problem of central configurations of the n-body problem with the general homogenous potential 1/rα, where α is a positive integer. A configuration q = (q1, q2, ⋅⋅⋅, qn) is called a super central configuration if there exists a positive mass vector m = (m1, ⋅⋅⋅, mn) such that q is a central configuration for m with mi attached to qi and q is also a central configuration for m′, where m′≠mandm′ is a permutation of m. The main result in this paper is the existence and classifications of super central configurations in the rectilinear three-body problem with general homogenous potential. Our results extend the previous work [Xie, Z., J. Math. Phys. 51, 042902 (2010)]10.1063/1.3345125 from the case in which α = 2 to the case in which α is a positive integer. Descartes’ rule of sign is extensively used in the proof of the main theorem.