Factorization of the Stability Polynomials of Ring Systems

TL;DR

This paper analyzes the stability of ring systems with dihedral symmetry by factorizing their stability polynomials using group representation theory, extending classical and recent results in the field.

Contribution

It introduces a systematic method to factorize stability polynomials of symmetric ring systems leveraging finite group representation theory, generalizing previous classical and modern findings.

Findings

Generalized factorization of stability polynomials for $D_n$-symmetric ring systems

Extended classical results from Maxwell to contemporary research

Provided a framework for stability analysis of ring systems in celestial mechanics

Abstract

Let $D_n$ be the dihedral group with $2n$ elements, and suppose $n$ is greater than one. We call ring system a finite $D_n$-symmetric set of points in $\mathbb{R}^2$. Ring systems have been used as models for planets surrounded by rings, and may be seen as relative equilibria of the $N$-body or the $N$-vortex problem. As a first significant step towards linear stability analysis, we study the factorization of the stability polynomial of an arbitrary ring system by systematically exploiting the ring's symmetry through representation theory of finite groups. Our results generalize contributions by J. C. Maxwell from mid-XIX century until contemporary authors such as J. Palmore and R. Moeckel, among others.