SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 181-207, January 2020.
We derive the exact equations of motion of the Keplerian dumbbell (KD) system, which consists of two point masses connected by a rigid massless rod, moving under the gravitational influence of a third point mass. We find several classes of continuous families of periodic orbits (principal periodic orbits) of this system, and we analyze their stability. In particular, these principal periodic orbits are stable both when the dumbbell arm points to the primary body and for a large circular orbital radius. At all times, this class of stable orbits has one of the dumbbell masses closer to the primary body, implying that axisymmetric bodies captured in these orbits have a hidden face relative to the primary body. This is an example of synchronous rotation without tidal locking. Moreover, we show that periodic points of the Poincaré map of the equations of motion do not necessarily correspond to periodic orbits of the dumbbell, which is a consequence of one of the symmetries of the Hamiltonian of the KD system.

中文翻译：

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开普勒哑铃系统的主要周期轨道

SIAM应用动力系统杂志，第19卷，第1期，第181-207页，2020年1月。
我们推导了开普勒哑铃（KD）系统的精确运动方程，该系统由通过刚性无质量杆连接的两个点质量组成，它们在第三点质量的重力影响下运动。我们发现了该系统的几类连续的周期性轨道族（主要周期性轨道），并分析了它们的稳定性。特别是，当哑铃臂指向主体时，以及对于较大的圆形轨道半径，这些主要周期性轨道都是稳定的。在任何时候，这类稳定的轨道都有一个更接近于主体的哑铃质量，这意味着捕获在这些轨道中的轴对称物体相对于主体具有隐藏的面。这是没有潮汐锁定的同步旋转的示例。此外，