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arXiv · 1707.05937

Numerical Methods and Closed Orbits in the Kepler-Heisenberg Problem

Victor Dods·Corey Shanbrom
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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.

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Victor Dods, Corey Shanbrom. 2017-07-19. Numerical Methods and Closed Orbits in the Kepler-Heisenberg Problem. https://doi.org/10.1080/10586458.2017.1416709

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