Linear Stability for Some Symmetric Periodic Simultaneous Binary Collision Orbits in the Four-Body Problem

TL;DR

This paper analyzes the linear stability of specific symmetric periodic orbits in four-body problems, confirming previous numerical results and extending understanding of stability in symmetric configurations.

Contribution

It applies an analytic-numerical method to determine stability of symmetric periodic orbits in collinear and planar four-body problems, verifying and extending prior numerical findings.

Findings

Confirmed linear stability of certain symmetric orbits in collinear four-body problem

Validated earlier numerical stability results through analytic-numerical methods

Extended stability analysis to symmetric planar four-body problem

Abstract

We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four body problem with masses 1, m, m, 1, and also in a symmetric planar four-body problem with equal masses. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability.