Communications in Mathematical Physics

, Volume 345, Issue 3, pp 797–879 | Cite as

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

  • Jinxin Xue
  • Dmitry Dolgopyat


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 Q 1 and Q 2 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 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.


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Supplementary material

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA
  2. 2.University of MarylandCollege ParkUSA

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