Periodic solutions to a forced Kepler problem in the plane

Abstract

Given a smooth function $U(t,x)$, $T$-periodic in the first variable and satisfying $U(t,x)=\mathcal{O}(|x{|}^{\alpha })$ for some $\alpha \in (0,2)$ as $|x|\to \infty$, we prove that the forced Kepler problem $$\ddot{x}=-\frac{x}{|x{|}^3}+{\nabla }_{x}U(t,x),x\in {\mathbb{R}}^2,$$ 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.