Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem

Tiancheng Ouyang; Zhifu Xie

doi:10.3934/dcds.2009.24.909

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2009, Volume 24, Issue 3: 909-932. Doi: 10.3934/dcds.2009.24.909

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Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem

Tiancheng Ouyang^1, and
Zhifu Xie^2,

1.
Department of Mathematics, Brigham Young University, Provo, Utah 84602
2.
Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806

Received: February 2008

Revised: December 2008

Published: August 2009

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Abstract

The purpose of this paper is to analyze the asymptotic properties of collision orbits of Newtonian $N$-body problems. We construct new coordinates and time transformation that regularize the singularities of simultaneous binary collisions in the collinear four-body problem. The motion in the new coordinates and time scale across simultaneous binary collisions at least $C^2$. The explicit formulae are given in detail for the transformations and the extension of solutions. Furthermore, we study the behaviors of the motion approaching, across and after the simultaneous binary collision. Numerical simulations have been conducted for the special case in which the bodies are distributed symmetrically about the center of mass.

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Mathematics Subject Classification: 37N05, 70F10, 70F16, 34C60.

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