具有两个无穷小质量的四体问题椭圆相对平衡的线性稳定性
Linear stability of elliptic relative equilibria of four-body problem with two infinitesimal masses
AI总结：
针对两个无穷小质量趋向同一拉格朗日点的四体问题，本文分析Xia给出的四个中心构型，证明非凸构型对应的椭圆相对平衡恒线性不稳定，并给出凸构型情形的参数稳定性条件。
摘要：
本文考虑具有两个无穷小质量的四体问题的椭圆相对平衡。最有趣的情形是两个小质量趋向同一个拉格朗日点$L_4$（或$L_5$）。在文献\cite{Xia}中，Z. Xia证明了存在四个中心构型：其中两个是非凸的，另外两个是凸的。我们证明，由非凸中心构型产生的椭圆相对平衡总是线性不稳定的；而对于由凸中心构型产生的椭圆相对平衡，则给出了其关于参数的线性稳定性条件。
英文摘要：
In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case is when the two small masses tend to the same Lagrangian point $L_4$ (or $L_5$). In \cite{Xia}, Z. Xia showed that there exist four central configurations: two of them are non-convex, and the other two are convex. We prove that the elliptic relative equilibria raised from the non-convex central configurations are always linearly unstable; while for the elliptic relative equilibria raised from the convex central configurations, the conditions of linear stability with respect to the parameters are given.
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