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Inverse problem of central configurations and singular curve in the collinear 4-body problem

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Abstract

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S m (q) be the permutational admissible set about m = (m 1, m 2, . . . , m n ) denoted

$$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$$

The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:

  1. (1)

    If \({m\in S(q)}\), then either # S m (q) = 0 or # S m (q) = 1.

  2. (2)

    #S m (q) = 1 only in the following cases:

    1. (i)

      If s = t, then S m (q) = {(m 4, m 3, m 2, m 1)}.

    2. (ii)

      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S m (q) = {(m 2, m 4, m 1, m 3)} or S m (q) = {(m 3, m 1, m 4, m 2)}.

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  1. Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA, 23806, USA

    Zhifu Xie

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  1. Zhifu Xie
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Correspondence to Zhifu Xie.

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Partially supported by RIG Grant (code 2137) 2008–2009 from Virginia State University.

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Xie, Z. Inverse problem of central configurations and singular curve in the collinear 4-body problem. Celest Mech Dyn Astr 107, 353–376 (2010). https://doi.org/10.1007/s10569-010-9280-8

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