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 a $ n $ type semiconductor with, and is known to be a
non-integrable Hamiltonian system. Many approximate periodic
solutions have been found by numerical calculation \cite{numerical-solution},
but they have not 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. It
is shown that the collision solution does not become the minimizer by
constructing an orbit with an action functional smaller than this
collision orbit by local deformation. This holds for any mu \in
(0,1). Reversibility allows the periodic orbit to be constructed from
the obtained minimizer.