Generalized periodic orbits in some restricted three-body problems (Q2021508)

The authors treat the circular and elliptic restricted three-body problems in inertial frames as periodically forced Kepler problems with additional singularities. Their model is a Kepler problem with an external periodic force. It is a classical problem in the theory of perturbations looking for periodic solutions for small eccentricity \(e\). In typical situations the periodic orbits are found in the neighbourhood of a prescribed closed orbit of the unperturbed Kepler problem and they have no collisions. For the circular problem it is considered in a fixed inertial frame in the 3D space, and not in a rotating frame. The authors apply this to show that an arbitrarily large number of generalized \(T\)-periodic solutions of a general class of circular or elliptic restricted three-body problems with any eccentricity \(e\in [0,1)\) can be found when the mass ratio of the primaries is sufficiently small. These solutions may collide with the big primary and they found periodic motions of the small body in a neighbourhood of this big primary in the 3D space.