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. 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. Reversibility allows the periodic orbit to be constructed from the minimizer obtained via the action functional.

The Hill problem models the motion of a particle near a planet. In this paper, we show the existence of symmetric periodic orbits in the spatial Hill problem by using the variational method. We also study the problem under a constraint on a prescribed plane and show the existence of periodic orbits in the problem. The obtained orbits are applicable to artificial satellites around the Earth.