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arXiv · 1806.01641

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

Bowen Liu
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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 $(u,e)\in (1/\sqrt{3}, \sqrt{3}) \times [0,1)$.

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Bowen Liu. 2018-06-02. Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem. https://doi.org/10.1088/1361-6544/ac288e

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