Periodic orbits in the kepler-heisenberg problem

Abstract

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with k-fold rotational symmetry for any odd integer κ ≥ 3. Approximations are shown for κ = 3. © American Institute of Mathematical Sciences.