Noncollision Singularities in Planar Two-center-two-body Problem

Noncollision Singularities in Planar Two-center-two-body Problem

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Jinxin Xue, Institute for Advanced Study
Fine Hall 601

In this work we study a model called planar 2-center-2-body 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. 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 construction 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 four-body problem case of the Painleve conjecture.This is a joint work with Dmitry Dolgopyat.