On the Minimizing Triple Collision Orbits in the Planar Newtonian Three-Body Problem

In this paper, we study the minimizing triple collision orbits in the planar Newtonian three-body problem with arbitrary masses. We show that for a given non-collinear initial configuration, the minimizing triple collision orbit is collision-free until a simultaneous collision, and its limiting configuration is the Lagrangian configuration with the same orientation as the initial configuration. For the collinear initial configuration, under a certain technical assumption, there exist two minimizing orbits. The limiting configurations of these orbits are the two opposite Lagrangian configurations.