Numerical Methods and Closed Orbits in the Kepler–Heisenberg Problem

Abstract

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 parametrized by the orbit's angular momentum. We provide links to all code developed.

An introduction to the Kepler-Heisenberg problem

“…All code is hosted publicly as free, open-source software; explicit instructions to reproduce each plot are provided there, with all relevant documentation and licensing. All results in [3] are thus 100% available and exactly reproducible. Our Python program searches for a closed orbit nearby some given initial condition, which may be input directly or randomly generated.…”

“…However, the proof followed the direct method in the calculus of variations and provided very little information about such orbits beyond their existence. Numerical methods in [3] allowed the discovery of a very rich and beautiful symmetry structure.…”