Numerical Methods and Closed Orbits in the Kepler–Heisenberg Problem

ABSTRACT

The Kepler–Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. This system is known to admit closed orbits, which all lie within a fundamental integrable subsystem. Here, we develop a computer program which finds these closed orbits using Monte Carlo optimization with a shooting method, and applying a recently developed symplectic integrator for nonseparable Hamiltonians. Our main result is the discovery of a family of flower-like periodic orbits with previously unknown symmetry types. We encode these symmetry types as rational numbers and provide evidence that these periodic orbits densely populate a one-dimensional set of initial conditions parameterized by the orbit's angular momentum. We provide links to all code developed.

Keywords:

• Periodic orbits
• kepler problem
• heisenberg group
• fundamental solution to laplacian
• symplectic integrator

Mathematics Subject Classification:

• 65P10
• 70H12
• 70F05
• 53C17

Acknowledgments

The authors are very grateful to the referee for suggestions which significantly improved the exposition and focus of this article.

Notes

1 Thanks to Michael VanValkenburgh (Sac State) for correcting the value α = 1/8π, which incorrectly appeared as 2/π in a previous paper; no prior results are affected by this change.

Additional information

Funding

American Mathematical Society (Simons Travel Grant)

National Science Foundation (DMS-1440140)