Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory

TL;DR

This paper introduces a new analytical method using index theory to determine the linear stability of elliptic Lagrangian solutions in the planar three-body problem across the full parameter range, improving upon previous perturbation and numerical approaches.

Contribution

The authors develop a rigorous analytical framework based on $oldsymbol{ heta}$-index theory and linear operators to analyze stability in the full parameter space, identifying key degeneracy curves and stability transitions.

Findings

Identified three critical curves in the parameter space affecting stability.

Proved stability changes occur when crossing these degeneracy curves.

Analyzed the stability behavior as eccentricity approaches 1.

Abstract

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]\times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(\bb,e)$ range $[0,9]\times [0,1)$ via the $\om$-index theory of symplectic paths for $\om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $\om$-index decreasing property of the solutions in $\bb$ for fixed $e\in [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]\times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $\om$-degeneracy curves for $\om\not=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(\bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.