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

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.