Number of central configurations and singular surfaces in the mass space in the collinear four-body problem

Abstract:

For a given $m=(m_1,m_2,\cdots , m_n)\in (\mathbf {R}^+)^n$, let $p$ and $q\in (\mathbf {R}^d)^n$ be two central configurations for $m$. Then we call $p$ and $q$ geometrically equivalent and write $p\sim q$ if they differ by a rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by $L(n,m)$ the set of geometric equivalence classes of $n$-body collinear central configurations for any given mass vector $m$. There are other different understandings of equivalence of central configurations in the collinear $n$-body problem. Under the usual definition of equivalence of central configurations in history, permutations of the bodies are not allowed, and we call them permutation equivalence. In this case Euler found three collinear central configurations and Moulton generalized to $n!/2$ central configurations for any given mass $m$ in the collinear $n$-body problem under permutation equivalence. In particular, the number of central configurations becomes from 12 under permutation equivalence to 1 under geometric equivalence for four equal masses in the collinear four-body problem. The main result in this paper is the discovery of the explicit parametric expressions of the union $H_4$ of the singular surfaces in the mass space $m=(m_1,m_2,m_3,m_4)\in$ $(\mathbf {R}^+)^4$, which decrease the number of collinear central configurations under geometric equivalence. We prove that the number of central configurations $^\#L(4,m)=4!/2-1=11$ if $m_1, m_2, m_3$ and $m_4$ are mutually distinct and $m\in H_4$.