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.Comment: 9 pages, 7 figures, completed in residence at MSRI; updated all
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