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

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August 2009, 24(3): 909-932. doi: 10.3934/dcds.2009.24.909

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

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

Received February 2008 Revised December 2008 Published April 2009

Abstract
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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.

Keywords: Collinear Four-Body Problem, Single Binary Collision, N-body problem, Solution with Collisions., Simultaneous Binary Collision, Regularization of Singularity.

Mathematics Subject Classification: 37N05, 70F10, 70F16, 34C6.

Citation: Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909

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