Periodic perturbations of central force problems and an application to a restricted $3$-body problem

Abstract

We consider a perturbation of a central force problem of the form \begin{equation*} \ddot x = V'(|x|) \frac{x}{|x|} + \varepsilon \,\nabla_x U(t,x), \quad x \in \mathbb{R}^{2} \setminus \{0\}, \end{equation*} where $\varepsilon \in \mathbb{R}$ is a small parameter, $V\colon (0,+\infty) \to \mathbb{R}$ and $U\colon \mathbb{R} \times (\mathbb{R}^{2} \setminus \{0\}) \to \mathbb{R}$ are smooth functions, and $U$ is $\tau$-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem ($\varepsilon=0$) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincar\'{e}-Birkhoff fixed point theorem to prove the existence of non-circular $\tau$-periodic solutions bifurcating from invariant tori at $\varepsilon=0$. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential $V(r)=\kappa/r^{\alpha}$ for $\alpha\in(-\infty,2)\setminus\{-2,0,1\}$). Finally, an application is given to a restricted $3$-body problem with a non-Newtonian interaction.