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

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Abstract

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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Authors and Affiliations

  1. University of Chicago, Chicago, IL, 60637, USA

    Jinxin Xue

  2. University of Maryland, College Park, MD, 20740, USA

    Dmitry Dolgopyat

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  1. Jinxin Xue
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  2. Dmitry Dolgopyat
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Correspondence to Jinxin Xue.

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Communicated by K. Khanin

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Xue, J., Dolgopyat, D. Non-Collision Singularities in the Planar Two-Center-Two-Body Problem. Commun. Math. Phys. 345, 797–879 (2016). https://doi.org/10.1007/s00220-016-2688-6

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  • DOI: https://doi.org/10.1007/s00220-016-2688-6

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