Action-Angle Variables in a Particular Case of the Restricted Three-Body Problem

摘要

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.