Eulerian collinear configuration for 3-body problem

TL;DR

This paper proves the existence of a special non-collision trajectory in the planar 3-body problem that maintains an Eulerian collinear configuration at all times, expanding understanding of possible configurations beyond classical solutions.

Contribution

It demonstrates the existence of a non-minimizing, non-collision Eulerian collinear configuration trajectory without winding number restrictions in the planar 3-body problem.

Findings

Existence of a non-collision Eulerian collinear trajectory at all times.

The trajectory is not a variational minimizer of the Lagrangian action.

No restriction on the winding number of the configuration.

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 $deg(q_i-q_j)\neq0 \,(i\neq j)$.