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$\ddot{x}=V^{\prime }\left(\right|x\left|\right)\frac{x}{\left|x\right|}+\epsilon {\nabla }_{x}U(t,x),x\in R^2\setminus \left\{0\right\},$ where $\epsilon \in R$ is a small parameter, $V:(0,+\infty )\to R$ and $U:R\times (R^2\setminus \{0\left\}\right)\to R$ are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem ($\epsilon =0$) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at $\epsilon =0$. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential $V\left(r\right)=\kappa /r^{\alpha }$ 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.