Collinear Central Configuration in Four-Body Problem

Abstract

In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration. In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central.

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

“…Lemma 4.1 (Ouyang and Xie 2005 Theorem 2). Let q = (q 1 , q 2 , q 3 , q 4 ) = (−s − 1, −1, 1, t + 1) be a configuration in the collinear four-body problem.…”

“…Ouyang and Xie (2005) (pp. 151 and 152) found explicitly the unique solution of masses to the central configuration Eq.…”

“…The notations and definitions in this paper are similar to those in Long and Sun (2004), Ouyang and Xie (2005) and Xie (2010). For any natural number n ∈ N, we denote by P(n) the set of all permutations of {1, 2, .…”

“…Albouy and Moeckel (2000) proved that a given configuration determines a two-parameter family of masses making it central where masses are allowed to be negative. Ouyang and Xie (2005) found the region for collinear 4-body configurations for which there exists a positive mass vector making it central. Long andSun (2004, 2002a) and Xie (2010) studied the collinear 3-body problem and some general properties in the n-body problem.…”

Central configurations in the collinear 5-body problem

“…Ouyang and Xie in [6] of central configurations of collinear 4-bodies and identified possible conditions to choose positive masses while maintaining a central configuration. The authors established an expression for the 4 masses depending on the position x and the center of mass u, which give central configurations in the collinear 4-body problem.…”