一个曲边Sitnikov问题的稳定性交换
Stability interchanges in a curved Sitnikov problem
摘要：
我们考虑一个弯曲的Sitnikov问题，其中一个无穷小的粒子在一个圆上运动的引力影响下的两个相等的质量在开普勒运动在一个平面内垂直于该圆。有两个平衡点，我们研究它们的稳定性。我们表明，其中一个平衡点进行稳定的交换作为半长轴的开普勒椭圆接近该圆的直径。为了得到这个结果，我们首先制定和证明稳定性交换的一般定理，然后我们将其应用到我们的模型。我们的模型的动机与常曲率空间中的$n$体问题有关。
英文摘要：
We consider a curved Sitnikov problem, in which an infinitesimal particle moves on a circle under the gravitational influence of two equal masses in Keplerian motion within a plane perpendicular to that circle. There are two equilibrium points, whose stability we are studying. We show that one of the equilibrium points undergoes stability interchanges as the semi-major axis of the Keplerian ellipses approaches the diameter of that circle. To derive this result, we first formulate and prove a general theorem on stability interchanges, and then we apply it to our model. The motivation for our model resides with the $n$-body problem in spaces of constant curvature.
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