Linear motions in a periodically forced Kepler problem (Q555196)

The one-dimensional Kepler problem with a periodic forcing term has at most one classical periodic solution and very simple dynamics. The aim of this paper is to show that, when collisions are allowed, a rich dynamics of twist type arises around the singularity. Bouncing solutions are defined in a natural way by continuing the energy at a collision. Then a suitable Poincaré section of the flux is well-defined and symplectic. The main result proves the existence and multiplicity of periodic and subharmonic solutions by using an elementary version of the Poincaré-Birkhoff Theorem. As the author remarks, it may be expected that the application of more sophisticated versions of the Poincaré-Birkhoff Theorem or KAM and Aubry-Mather Theory could lead to a more precise picture of the underlying dynamics, including stability of periodic solutions, boundedness and complex recurrence.