The golden ratio and super central configurations of the n-body problem

In this paper, we consider the problem of central configurations of the n-body problem with the general homogeneous potential 1/rα. 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^′≠m and m^′ is a permutation of m. The main discovery in this paper is that super central configurations of the n-body problem have surprising connections with the golden ratio φ. Let r be the ratio of the collinear three-body problem with the ordered positions q₁, q₂, q₃ on a line. q is a super central configuration if and only if 1/r₁(α)<r<r₁(α) and r≠1, where r₁(α)>1 is a continuous function such that , the golden ratio. The existence and classification of super central configurations are established in the collinear three-body problem with general homogeneous potential 1/rα. Super central configurations play an important role in counting the number of central configurations for a given mass vector which may decrease the number of central configurations under geometric equivalence.