Exotic Hill Problem: Hall motions and symmetries

TL;DR

This paper extends Hill's equations to non-commutative particles, revealing that at a critical angular velocity, motions follow the Hall law and symmetries form Heisenberg algebras, with symmetry reduction at the critical point.

Contribution

It introduces non-commutative particles into the Hill problem and analyzes the resulting symmetries and motions at critical angular velocities.

Findings

At a critical angular velocity, particles follow the Hall law.

Symmetries form two Heisenberg algebras, reducing to one at the critical point.

The system exhibits exotic non-commutative behavior in celestial mechanics.

Abstract

Our previous study of a system of bodies assumed to move along almost circular orbits around a central mass, approximately described by Hill's equations, is extended to "exotic" [alias non-commutative] particles. For a certain critical value of the angular velocity, the only allowed motions follow the Hall law. Translations and generalized boosts span two independent Heisenberg algebras with different central parameters. In the critical case, the symmetry reduces to a single Heisenberg algebra.