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)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.