Linear stability of the Lagrangian triangle solutions for quasihomogeneous potentials

TL;DR

This paper investigates the linear stability of Lagrangian triangle solutions in n-body problems with homogeneous and quasihomogeneous potentials, generalizing classical results and identifying conditions for stability based on potential degree and configuration size.

Contribution

It extends classical stability results to quasihomogeneous potentials and provides new criteria for stability depending on potential parameters and configuration size.

Findings

Relative equilibria with degree a>2 are spectrally unstable.

Classical Routh stability criterion is recovered for homogeneous potentials.

New stability conditions are derived for quasihomogeneous potentials depending on mass and size.

Abstract

In this paper we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. Firstly, in the case the potential is a homogeneous function of degree $-a$, we find that any relative equilibrium of the $n$-body problem with $a>2$ is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.