Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

TL;DR

This paper demonstrates that the linear stability analysis of elliptic triangle solutions in the charged three-body problem can be directly related to that of the classical three-body problem, enabling the application of existing stability results.

Contribution

It establishes a transformation linking the linearized systems of charged and classical three-body problems, allowing known stability results to be applied to the charged case.

Findings

Linearized system of charged three-body problem can be transformed to classical case.

Existing stability results are applicable to charged three-body solutions.

Provides a method to analyze stability of charged three-body configurations.

Abstract

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Mart\'{\i}nez, Sam\`{a} and Sim\'{o} ([15] in J. Diff. Equa.) of 2006 and results of Hu, Long and Sun ([6] in Arch. Ration. Mech.Anal.) of 2014 can be applied to these solutions of the charged three-body problem to get their linear stability.