Non-Collision Singularities in the Planar Two-Center-Two-Body Problem

In this paper, we study a restricted four-body problem called the planar two-center-two-body problem. In the plane, we have two fixed centers Q₁ and Q₂ of masses 1, and two moving bodies Q₃ and Q₄ of masses μ≪1. They interact via Newtonian potential. Q₃ is captured by Q₂, and Q₄ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions that lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all earlier collisions. This problem is a simplified model for the planar four-body problem case of the Painlevé conjecture.