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/r(alpha). 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(i) with nit attached to q(i) and q is also a central configuration for m', where m' not equal 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 phi. Let r be the ratio vertical bar q(3)-q(2)vertical bar/vertical bar q(2)-q(1)vertical bar 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) (alpha) < r < r(1) (alpha) and r not equal 1, where r(1) (alpha) > 1 is a continuous function such that lim(alpha -> 0) r(1)(alpha) = phi, 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(alpha). 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. (C) 2011 Elsevier Inc. All rights reserved.