An introduction to the Kepler-Heisenberg problem

TL;DR

This paper introduces the Kepler-Heisenberg problem, a novel dynamical system combining classical Kepler dynamics with sub-Riemannian geometry, highlighting its intriguing properties and open questions.

Contribution

It provides an overview of the known and unknown aspects of the Kepler-Heisenberg problem, framing it as a new research direction in dynamical systems.

Findings

The system exhibits rich and complex behavior.

It presents numerous open and tractable questions.

The problem bridges classical mechanics and sub-Riemannian geometry.

Abstract

Here we provide an overview of what is known, and what is not known, about an interesting dynamical system known as the Kepler-Heisenberg problem. The main idea is to pose a version of the classical Kepler problem of planetary motion, but in a sub-Riemannian setting. The result is system which is surprisingly rich and beautiful, mysterious in some ways but tame in others, offering a substantial number of questions which seem non-trivial yet tractable.