Self-similarity in the Kepler-Heisenberg problem

TL;DR

This paper studies the self-similarity of zero-energy orbits in the Kepler-Heisenberg problem, revealing they form three distinct families and exhibit finite-time collisions under certain conditions.

Contribution

It demonstrates that zero-energy orbits in the Kepler-Heisenberg problem are self-similar and stratify into three families, providing new insights into their structure and collision behavior.

Findings

Zero-energy orbits are self-similar under mild conditions.

Orbits stratify into future collision, past collision, and quasi-periodic families.

Collisions, if they occur, happen in finite time.

Abstract

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.