Variational proof of the existence of periodic orbits in the anisotropic Kepler problem

The anisotropic Kepler problem is a model of the motion of free electrons on an n-type semiconductor and is known to be a non-integrable Hamiltonian system. While many approximate periodic solutions have been found through numerical calculation (Sumiya et al. in Artif Life Robot 19:262–269, 2014), none have been rigorously proved to exist. In this paper, we first show that the action functional of the anisotropic Kepler problem has a minimizer under a fixed region condition with boundary conditions on a vertical half-line. Next, we identify the smallest collision trajectory that satisfies the same boundary conditions. By constructing an orbit with an action functional smaller than this collision orbit via local deformation, we show that the collision solution does not become the minimizer. This holds for any μ∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu \in (0,1) $$\end{document}. Reversibility allows the periodic orbit to be constructed from the minimizer obtained via the action functional.