Dynamics of the the dihedral four-body problem

TL;DR

This paper analyzes the dynamics of four equal-mass particles constrained by dihedral symmetry under a homogeneous potential, focusing on collision behavior using McGehee coordinates.

Contribution

It provides a detailed qualitative analysis of collision orbits in a symmetric four-body problem with zero angular momentum.

Findings

Characterization of flow near total collision

Identification of collision orbit behaviors

Insights into symmetry constraints on dynamics

Abstract

Consider four point particles with equal masses in the euclidean space, subject to the following symmetry constraint: at each instant they are symmetric with respect to the dihedral group $D_2$, that is the group generated by two rotations of angle $\pi$ around two orthogonal axes. Under a homogeneous potential of degree $-\alpha$ for $0<\alpha<2$, this is a subproblem of the four-body problem, in which all orbits have zero angular momentum and the configuration space is three-dimensional. In this paper we study the flow in McGehee coordinates on the collision manifold, and discuss the qualitative behavior of orbits which reach or come close to a total collision.