We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. These masses always move in rhomboidal CC with diagonals and . The sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. The masses of the primaries are and . The masses and are written as functions of parameters and such that they always form a rhomboidal central configuration. The evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle and identify the value of Jacobian constant for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for , there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable.

中文翻译：

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菱形受限六体问题的稳定性分析

我们讨论受限菱形六体问题（RR6BP），它在菱形的顶点有四个正质量，第五个质量在两条对角线的交点处。这些质量总是在菱形 CC 中移动，对角线和。第六个物体的质量非常小，不会影响五个物体（也称为初级物体）的运动。初选的群众是 和 . 群众和被写为参数的函数和，使得它们总是形成一个菱形中央配置。针对正质量的固定值讨论了零速度曲线的演变。使用第一次运动积分，我们推导出测试粒子可能运动的区域，并确定这些区域断开的不同能量区间的雅可比常数值。使用半解析技术，我们展示了轴上和轴外平衡解的存在性和唯一性。我们证明，对于，总有 12 个平衡点。我们还表明所有 12 个平衡点都是不稳定的。