
Mathematics 2013
NonCollision singularities in the Planar twoCentertwoBody problemAbstract: In this paper, we study a model of simplified fourbody problem called planar twocentertwobody problem. In the plane, we have two fixed centers $Q_1=(\chi,0)$, $Q_2=(0,0)$ of masses 1, and two moving bodies $Q_3$ and $Q_4$ of masses $\mu\ll 1$. They interact via Newtonian potential. $Q_3$ is captured by $Q_2$, and $Q_4$ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar fourbody problem case of the Painlev\'{e} conjecture.
