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

Abstract

In this paper, we consider the problem of central configurations of the n-body problem with the general homogeneous potential 1 / r α . A configuration q = ( q 1 , q 2 , … , q n ) is called a super central configuration if there exists a positive mass vector m = ( m 1 , … , m n ) such that q is a central configuration for m with m i attached to q i 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 | q 3 − q 2 | | q 2 − q 1 | of the collinear three-body problem with the ordered positions q 1 , q 2 , q 3 on a line. q is a super central configuration if and only if 1 / r 1 ( α ) < r < r 1 ( α ) and r ≠ 1 , where r 1 ( α ) > 1 is a continuous function such that lim α → 0 r 1 ( α ) = φ , the golden ratio. The existence and classification of super central configurations are established in the collinear three-body problem with general homogeneous potential 1 / r α . 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.