Linear instability of elliptic rhombus solutions to the planar four-body problem

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