Eulerian collinear configuration for 3-body problem

Abstract

For 3-body problems with any given masses m 1 , m 2 , m 3 > 0, there exist only Eluerian collinear central configuration and Lagrangian equilateral-triangle central configuration. In 2004, for planar 3-body problem, Zhang and Zhou (Celestial Mech. Dyn. Astron. 90: 239-243, 2004) proved that the variational minimizer of the Lagrangian action restricted on a suitable loop space, is just an Eulerian collinear central configuration. In this paper, for spatial 3-body problem, we prove that there exists other trajectory q, not the variational minimizer of the Lagrangian action, is also an Eulerian collinear central configuration. Moreover, we do not need the restriction on the winding number deg(q i − q j ) = 0 (i = j).Keywords 3-body problem; Eulerian collinear configuration; Mountain pass theorem MSC (2010) 70F07, 70F15