Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem

TL;DR

This paper investigates the linear stability of elliptic rhombus solutions in the planar four-body problem, proving instability under certain shape and eccentricity conditions using advanced mathematical tools.

Contribution

It provides the first analytical proof of linear instability for elliptic rhombus solutions across a broad parameter range, extending previous numerical findings.

Findings

Proves linear instability for specific shape and eccentricity ranges.

Extends instability results to a wider parameter space.

Uses $ ext{-}Maslov$ index theory and trace formula for analysis.

Abstract

In this paper, we study the linear stability of the elliptic rhombus solutions, which are the Keplerian homographic solution with the rhombus central configurations in the classical planar four-body problems. Using $\omega$-Maslov index theory and trace formula, we prove the linear instability of elliptic rhombus solutions if the shape parameter $u$ and the eccentricity of the elliptic orbit $e$ satisfy $(u,e) \in (1/\sqrt{3}, u_2)\times [0, \hat{f}(\frac{27}{4})^{-1/2})\cup (u_2, 1/u_2) \times [0,1)\cup ( 1/u_2, \sqrt{3})\times [0, \hat{f}(\frac{27}{4})^{-1/2})$ where $u_2\approx 0.6633$ and $\hat{f}(\frac{27}{4})^{-1/2} \approx 0.4454$. Motivated on numerical results of the linear stability to the elliptic Lagrangian solutions in [R. Mart\'{\i}nez, A. Sam\`{a}, and C. Sim\'{o}, J. Diff. Equa., 226(2006): 619--651.], we further analytically prove the linear instability of elliptic rhombus solutions for $(u,e)\in (1/\sqrt{3}, \sqrt{3}) \times [0,1)$.