Keplerian Dynamics on the Heisenberg Group and Elsewhere

TL;DR

This paper explores the formulation and solution of Kepler's problem within non-Euclidean geometries, specifically on the Heisenberg group, highlighting successes, failures, and open questions in extending classical mechanics to subRiemannian spaces.

Contribution

It demonstrates the first successful formulation and solution of Kepler's problem on the Heisenberg group, expanding the understanding of classical mechanics in non-Euclidean geometries.

Findings

Kepler's problem can be posed and solved on the Heisenberg group.

The problem cannot be posed on the rank two lattice.

Partial solutions are found on the integers.

Abstract

Posing Kepler's problem of motion around a fixed "sun" requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the "sun") provides the potential energy. Posing Kepler's three laws (with input from Galileo) requires symmetry conditions. The metric space must be homogeneous, isotropic, and admit dilations. Any Riemannian manifold enjoying these three symmetry properties is Euclidean. So if we want a semblance of Kepler's three laws to hold but also want to leave the Euclidean realm, we are forced out of the realm of Riemannian geometries. The Heisenberg group (a subRiemannian geometry) and lattices provide the simplest examples of metric spaces enjoying a semblance of all three of the Keplerian symmetries. We report success in posing, and solving, the Kepler problem on the Heisenberg group. We report failures in posing the Kepler problem on the rank two lattice and partial success in solving the problem on the integers. We pose a number of questions.