Hamiltonian solutions of the 3-body problem in (2+1) gravity

We present a full study of the 3-body problem in gravity in a flat (2+1)-dimensional spacetime, and in the nonrelativistic limit of small velocities. We provide an explicit form of the Arnowitt-Deser-Misner Hamiltonian in a regular coordinate system and we set up all the ingredients for canonical quantization. We emphasize the rle of a U(2) symmetry under which the Hamiltonian is invariant and which should generalize to a U(N - 1) symmetry for N bodies. This symmetry seems to stem from a braid group structure in the operations of looping of particles around each other, and guarantees the single valuedness of the Hamiltonian. Its rle for the construction of single-valued energy eigenfunctions is also discussed.