Linear Stability Analysis of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem

TL;DR

This study uses symmetry reduction to numerically analyze the linear stability of symmetric periodic orbits with binary collisions in a four-body problem, revealing stability changes as the mass parameter varies.

Contribution

It introduces a numerical method employing symmetry reduction to analyze stability of symmetric periodic orbits with binary collisions in the four-body problem.

Findings

Stability varies with mass parameter m.

Orbits are unstable near m=0.01.

Orbits are stable near m=1.

Abstract

We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four-body problem with a mass $m\in(0,1]$ as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a $3\times 3$ matrix for each $m\in(0,1]$ obtained from numerical integration of the linearized regularized equations along only the first one-eighth of each regularized periodic orbit. The results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as $m$ varies over $(0,1]$, with linear instability for $m$ close or equal to 0.01, and linear stability for $m$ close or equal to 1.