Periodic solutions and regularization of a Kepler problem with time-dependent perturbation

TL;DR

This paper proves the existence of multiple periodic solutions in a Kepler problem with small, time-periodic perturbations, allowing for collisions and using a generalized solution framework.

Contribution

It establishes the existence of arbitrarily many periodic solutions in perturbed Kepler problems with collisions, extending classical results to include generalized solutions.

Findings

At least N periodic solutions exist for any positive integer N.

Solutions can include collisions, handled via a generalized solution concept.

Results apply to both 2D and 3D Kepler problems with small perturbations.

Abstract

We consider a Kepler problem in dimension two or three, with a time-dependent $T$-periodic perturbation. We prove that for any prescribed positive integer $N$, there exist at least $N$ periodic solutions (with period $T$) as long as the perturbation is small enough. Here the solutions are understood in a general sense as they can have collisions. The concept of generalized solutions is defined intrinsically and it coincides with the notion obtained in Celestial Mechanics via the theory of regularization of collisions.