The two-dimensional three-body problem in the large magnetic field limit is integrable

•The 2-d, three charged particle problem in a strong magnetic field is shown to be integrable.•A spinorial description of the configuration of the system is presented.•The quantum mechanical problem is also discussed.

The problem of N   particles interacting through pairwise central forces is notoriously intractable for N≥3N≥3. Some remarkable specific cases have been solved in one dimension. Here we show that the guiding center approximation—valid for charges moving in two dimensions in the limit of large constant magnetic fields—simplifies the three-body problem for an arbitrary interparticle interaction invariant under rotations and translations, making it solvable by quadratures. A spinorial representation for the system is introduced, which allows a visualization of its phase space as the corresponding Bloch sphere. Finally, a discussion of the quantization of the problem is presented.