Abstract—

We have considered the case of a restricted three-body problem (material points), when the masses of the two main attracting bodies are equal. Their orbits have been assumed to be ellipses. The problem regarding the third body’s motion of negligible mass under the influence of the main body’s gravitational attraction (the Sitnikov problem) allows particular solutions for the following: the third body moves along a straight line passing through the center of mass of the main bodies and perpendicular to the plane of their orbits. We have assumed that the eccentricity of the main body orbits is small and we investigated the nonlinear problem of the existence of the third body’s periodic motion with a period multiple of the revolution period of the main bodies in their orbits. The problem of stability (in Lyapunov’s sense) of these periodic motions under both perturbations keeping the trajectory of the third body rectilinear and arbitrary spatial perturbations has also been solved.

中文翻译：

---

椭圆圆形Sitnikov问题中的次谐波振荡

摘要-

当两个主要吸引体的质量相等时，我们考虑了三体受限问题（物质点）的情况。他们的轨道被假定为椭圆形。关于第三物体在主体重力吸引的影响下可忽略的质量运动的问题（Sitnikov问题）为以下问题提供了特殊的解决方案：第三物体沿着直线移动，该直线穿过主体的质心，并且垂直于其轨道平面。我们假设主体轨道的偏心率很小，并且研究了存在于主体轨道旋转周期的周期倍数的第三主体的周期性运动存在的非线性问题。