Exact nonplanar solutions of the classical relativistic three-body problem

It was shown recently by Havas that for a Newtonian system of three particles there exists a class of exact solutions which is nonplanar, such that the particles revolve with constant angular velocity in circular orbits in parallel planes. Such motions are possible for central forces which are arbitrary functions of the mutual separations, provided that the forces are repulsive between one pair of particles and attractive between the two other pairs (which for Coulomb forces corresponds to two like charges and one unlike one). In this paper, it is established that such motions are possible also for relativistic three-body systems with time-symmetric electromagnetic or mesic interactions; they reduce to the Newtonian solutions in the appropriate limit. Beyond this, there also exist exact relativistic solutions with the same orbits which have no nonrelativistic counterpart. These include solutions with all electric charges of the same sign, as well as solutions with all three particles on the same side of the axis of rotation; degenerate cases include planar and nonplanar two-particle solutions and a one-particle circular solution.