Linear stability of homogeneous and quasi-homogeneous N-body problem by symmetry groups

TL;DR

This paper investigates the linear stability of specific N-body configurations, such as equilateral triangles and squares, using symmetry group methods to analyze the equations' linearized form.

Contribution

It applies symmetry group techniques to analyze the linear stability of relative equilibria in N-body problems with homogeneous and quasi-homogeneous potentials, extending previous methods.

Findings

Decomposition of stability matrices into 2x2 blocks for analysis.

Identification of stability conditions for specific configurations.

Extension of symmetry group methods to quasi-homogeneous potentials.

Abstract

Motivated by Xia-Zhou's recent work on applying symmetry groups to the N-body problem, we will study relative equilibria of the equilateral triangle and the square configurations under $\alpha$-homogeneous and quasi-homogeneous potentials with this method. After linearizing the corresponding second order equations, with appropriate coordinate transformations, we study the linear stability of the relative equilibria by decomposing each $2n\times 2n$ matrix into a series of $2\times 2$ matrices.