Periodic Solutions with Alternating Singularities in the Collinear Four-body Problem

TL;DR

This paper proves the existence of a symmetric periodic orbit with alternating binary and simultaneous binary collisions in the collinear four-body problem, using regularization and mathematical analysis.

Contribution

It introduces a new periodic solution with singularities in the four-body problem and demonstrates its existence through rigorous mathematical methods.

Findings

Existence of a symmetric periodic orbit with singularities.

Orbit features alternating binary and simultaneous binary collisions.

Methodology involves regularization and implicit function theorem.

Abstract

This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proven by using the implicit function theorem and a continuity argument on differential equations of the regularized Hamiltonian.