The two-dimensional three-body problem in a strong magnetic field is integrable

TL;DR

This paper demonstrates that the three-body problem in a strong magnetic field becomes integrable under the guiding center approximation, allowing solutions by quadratures for a broad class of interactions.

Contribution

It introduces a guiding center approximation that simplifies the three-body problem in two dimensions, making it exactly solvable for arbitrary rotationally and translationally invariant interactions.

Findings

The system is integrable and solvable by quadratures.

A spinorial representation visualizes phase space as a Bloch sphere.

Identification of Berry-Hannay rotational anholonomy.

Abstract

The problem of $N$ particles interacting through pairwise central forces is notoriously intractable for $N\geq3$. Some quite remarkable specific cases have been solved in one dimension, whereas higher-dimensional exactly solved systems involve velocity-dependent or many-body forces. Here we show that the guiding center approximation---valid for charges moving in two dimensions in a strong constant magnetic field---simplifies the three-body problem for an arbitrary interparticle interaction invariant under rotations and translations and makes it solvable by quadratures. This includes a broad variety of special cases, such as that of three particles interacting through arbitrary pairwise central potentials. A spinorial representation for the system is introduced, which allows a visualization of its phase space as the corresponding Bloch sphere as well as the identification of a Berry-Hannay rotational anholonomy. Finally, a brief discussion of the quantization of the problem is presented.