Simultaneous Binary Collisions for Collinear Four-Body Problem

TL;DR

This paper analytically studies singularities in collinear four-body problems, connecting solutions of decoupled and coupled cases through canonical transformations, providing a comprehensive understanding of simultaneous binary collisions.

Contribution

It introduces a method to analytically continue singularities and relates solutions between decoupled and coupled four-body problems using canonical transformations.

Findings

All solutions for the problem are identified.

The relationship between decoupled and coupled solutions is clarified.

Singularities are analytically continued using canonical transformations.

Abstract

In this paper, we use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions for the collinear four- body problem in both the decoupled case and the coupled case. All the solutions are found and more importantly, we describe the relationship between the decoupled solutions and the coupled solutions.