Abstract

The restricted problem of three bodies (material points) moving under the action of Newtonian gravitational attraction is considered. The masses of the main attracting points are equal, and the points themselves move in circular orbits around their common center of mass. The third point has a negligible mass and moves along a straight line perpendicular to the plane of orbits of the main points and passing through their center of mass. The problem about this motion is integrable. For the case where the third point motion is oscillatory, the procedure of introducing action–angle variables is proposed in this paper. Nonlinear oscillations of small amplitude and oscillations with arbitrarily large amplitudes in comparison with the distance between the main attracting points are considered as examples.

中文翻译：

---

受限三体问题的特殊情况下的动作角变量

摘要

考虑了三个物体（物质点）在牛顿引力作用下运动的受限问题。主要吸引点的质量相等，这些点本身围绕其共同的质心以圆形轨道运动。第三点的质量可以忽略不计，并且沿着垂直于主要点的轨道平面并穿过其重心的直线移动。关于这项议案的问题是可以解决的。对于第三点运动是振荡的情况，本文提出了引入作用角变量的程序。与主要吸引点之间的距离相比，小振幅的非线性振荡和大振幅的振荡被认为是示例。