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

Abstract

The purpose of this paper is to analyze the asymptotic properties ofcollision orbits of Newtonian $N$-body problems. We construct newcoordinates and time transformation that regularize thesingularities of simultaneous binary collisions in the collinearfour-body problem. The motion in the new coordinates and time scaleacross simultaneous binary collisions at least $C^2$. The explicitformulae are given in detail for the transformations and theextension of solutions. Furthermore, we study the behaviors of the motion approaching,across and after the simultaneous binary collision. Numericalsimulations have been conducted for the special case in which thebodies are distributed symmetrically about the center of mass.