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 $k\ge 3$. Approximations are shown for $k=3$.

19 pages, 3 figures

References in corpus (2)

• Small time heat kernel asymptotics at the sub-Riemannian cut locus
• Keplerian Dynamics on the Heisenberg Group and Elsewhere

Cited by in corpus (3)

• Self-similarity in the Kepler-Heisenberg problem
• Numerical Methods and Closed Orbits in the Kepler-Heisenberg Problem
• Non-integrability of the Kepler and the two-body problem on the Heisenberg group