Eulerian collinear configuration for 3-body problem

Abstract

For 3-body problem with any given masses $m_1, \,m_2,\,m_3>0$, there exist only Eulerian collinear central configuration and Lagrangian equilateral-triangle central configuration, and in this paper, for planar 3-body problem, we prove that there exists another non-collision trajectory $q$, which is not the variational minimizer of the Lagrangian action on the loop space $\overline{\Lambda_1}$, is also an Eulerian collinear central configuration at any instant. Moreover, we do not need the restriction condition on the winding number $deg(q_i-q_j)\neq0 \,(i\neq j)$.