# Periodic solutions to a forced Kepler problem in the plane

## TL;DR

This paper proves the existence of periodic solutions in a forced Kepler problem with time-dependent perturbations using variational methods, under certain growth conditions on the perturbation.

## Contribution

It establishes the existence of generalized periodic solutions for a class of forced Kepler problems with time-periodic perturbations, extending previous results to broader conditions.

## Key findings

- Existence of generalized T-periodic solutions proven
- Solutions exist under specific growth conditions on the perturbation
- Variational methods are effectively applied to this celestial mechanics problem

## Abstract

Given a smooth function $U(t,x)$, $T$-periodic in the first variable and satisfying $U(t,x) = \mathcal{O}(\vert x \vert^{\alpha})$ for some $\alpha \in (0,2)$ as $\vert x \vert \to \infty$, we prove that the forced Kepler problem $$ \ddot x = - \dfrac{x}{|x|^3} + \nabla_x U(t,x),\qquad x\in {\mathbb{R}}^2, $$ has a generalized $T$-periodic solution, according to the definition given in the paper [Boscaggin, Ortega, Zhao, \emph{Periodic solutions and regularization of a Kepler problem with time-dependent perturbation}, Trans. Amer. Math. Soc, 2018]. The proof relies on variational arguments.