Super central configurations of the n -body problem

In this paper, we consider the inverse problem of central configurations of the n -body problem. For a given q = ( q 1 , q 2 , … , q n ) ∊ ( R d ) n , let S ( q ) be the admissible set of masses by S ( q ) = { m = ( m 1 , … , m n ) ∣ m i ∊ R + , q is a central configuration for m } . For a given m ∊ S ( q ) , let S m ( q ) be the permutational admissible set about m = ( m 1 , m 2 , … , m n ) by S m ( q ) = { m ′ ∣ m ′ ∊ S ( q ) , m ′ ≠ m and m ′ is a permutation of m } . Here, q is called a super central configuration if there exists m such that S m ( q ) is nonempty. For any q in the planar four-body problem, q is not a super central configuration as an immediate consequence of a theorem proved by MacMillan and Bartky [“Permanent configurations in the problem of four bodies,” Trans. Am. Math. Soc. 34, 838 (1932)]. The main discovery in this paper is the existence of super central configurations in the collinear three-body problem. We proved that for any q in the collinear three-body problem and any m ∊ S ( q ) , S m ( q ) has at most one element and the detailed classification of S m ( q ) is provided.