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 Q 1 and Q 2 of masses 1, and two moving bodies Q 3 and Q 4 of masses μ ≪ 1 . They interact via Newtonian potential. Q 3 is captured by Q 2 , and Q 4 travels b...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 345; no. 3; pp. 797 - 879
Main Authors Xue, Jinxin, Dolgopyat, Dmitry
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2016
Subjects
Classical and Quantum Gravitation
Complex Systems
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
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Summary: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 μ ≪ 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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-016-2688-6