Generalized Periodic Orbits of Time-Periodically Forced Kepler Problem Accumulating the Center and of Circular and Elliptic Restricted Three-Body Problems

TL;DR

This paper proves the existence of infinitely many periodic orbits in time-periodically forced Kepler problems and related three-body problems, including orbits accumulating at the centers or primaries, even with collisions regularized.

Contribution

It establishes the existence of infinitely many periodic orbits in generalized forced Kepler and three-body problems using localization and persistence techniques.

Findings

Infinitely many periodic orbits exist in the considered systems.

Orbits can have double collisions with centers or primaries, which are regularized.

Orbits accumulate at the attractive centers or primaries.

Abstract

In this paper, we consider a time-periodically forced Kepler problem in any dimensions, with an external force which we only assume to be regular in a neighborhood of the attractive center. We prove that there exist infinitely many periodic orbits in this system, with possible double collisions with the center regularized, which accumulate the attractive center. The result is obtained via a localization argument combined with a result on $C^{1}$-persistence of closed orbits by a local homotopy-streching argument. Consequently, by formulating the circular and elliptic restricted three-body problems of any dimensions as time-periodically forced Kepler problems, we obtain that there exists infinitely many periodic orbits, with possible double collisions with the primaries regularized, accumulating to each of the primaries.