标题： 限制$4$体问题中椭圆相对平衡的线性稳定性：欧拉情况 显示英文标题

标题： Linear stability of the elliptic relative equilibria for the restricted $4$-body problem: the Euler case

摘要： 在本文中，我们考虑限制性$4$-体问题的椭圆相对平衡，其中三个主要天体形成一个欧拉共线配置，四个天体扩展为$\mathbf{R}^2$。我们得到了一般限制性$N$-体问题的辛约化。通过分析此限制性$4$-体问题与椭圆拉格朗日解之间的关系，我们通过$\omega$-Maslov指标获得了限制性$4$-体问题的线性稳定性。通过数值计算，我们也得到了对称情况下质量参数的稳定性条件。 显示英文摘要

摘要： In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic reduction to the general restricted $N$-body problem. By analyzing the relationship between this restricted $4$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $4$-body problem by the $\omega$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters for the symmetric cases.