平面四体问题椭圆菱形解的线性不稳定性
Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem
摘要：
本文研究了经典平面四体问题中具有菱形中心构型的开普勒单应解--椭圆菱形解的线性稳定性。利用$ω$指标理论和迹公式，证明了当椭圆轨道的形状参数$u$和偏心率$e$满足$(u,e) \in (1/\sqrt{3}, u_2)\times [0, \hat{f}(\frac{27}{4})^{-1/2})\cup (u_2, 1/u_2) \times [0,1)\cup ( 1/u_2, \sqrt{3})\times [0, \hat{f}(\frac{27}{4})^{-1/2})$时椭圆菱形解的线性不稳定性，其中$u_2\approx 0.6633$和$\hat{f}(\frac{27}{4})^{-1/2} \approx 0.4454$ .基于[R. Mart 'ınez，A. Samà，和C. Simó，J. Diff.等式，226（2006）：619- 651.]，我们进一步解析地证明了$(u,e)\in (1/\sqrt{3}, \sqrt{3}) \times [0,1)$椭圆菱形解的线性不稳定性。
英文摘要：
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 $ω$-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 $(u,e)\in (1/\sqrt{3}, \sqrt{3}) \times [0,1)$.
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