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

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\'ınez, A. Samà, and C. Simó, J. Diff. Equa., 226(2006): 619--651.], we further analytically prove the linear instability of elliptic rhombus solutions for $(u,e)\in (1/\sqrt{3},\sqrt{3})\times [0,1)$.

33 pages, 2 figures

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• Convex Four Body Central Configurations with Some Equal Masses
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Cited by in corpus (1)

• Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases