Numerical Methods and Closed Orbits in the Kepler-Heisenberg Problem

TL;DR

This paper develops a computational approach to find and analyze closed orbits in the Kepler-Heisenberg problem, revealing new flower-like periodic orbits with unique symmetries in a sub-Riemannian setting.

Contribution

It introduces a Monte Carlo shooting method with symplectic integration to discover new families of closed orbits in the Kepler-Heisenberg system, expanding understanding of its dynamical behavior.

Findings

Discovery of flower-like periodic orbits with novel symmetries

Periodic orbits densely populate a one-dimensional initial condition set

Provision of code for reproducibility and further research

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.