Linear Stability of Elliptic Rhombus Solutions of the Planar Four-body Problem

Bowen Liu

Published 2018-06-02Version 1

In this paper, we study the linear stability of the elliptic rhombus homographic solutions in the classical planar four-body problem which depends on the shape parameter $u \in (1/\sqrt{3}, \sqrt{3})$ and eccentricity $e\in [0,1)$. By an analytical result obtained in the study of the linear stability of elliptic Lagrangian solutions, we prove that the linearized Poincare map of elliptic rhombus solution possesses at least two pairs of hyperbolic eigenvalues, when $(u,e) \in (u_3, 1/u_3) \times [0,1)$ or $ (u,e) \in\l([1/\sqrt{3}, u_3) \cup(1/u_3, \sqrt{3}]\r) \times [0, \hat{f}(\frac{27}{4})^{-1/2})$ where $u_3\approx 0.6633$ and $\hat{f}(\frac{27}{4})^{-1/2} \approx 0.4454$. By a numerical result obtained in the study of the elliptic Lagrangian solutions, we analytically prove that the elliptic rhombus solution is hyperbolic, i.e., it possesses four pairs of hyperbolic eigenvalues, when $(u,e)\in [1/\sqrt{3}, \sqrt{3}] \times [0,1)$.