The three-body problem on the sphere and its reversing symmetries

In this work, we establish the concept of reversing symmetry in the three-body problem on the sphere, a novel approach that has not been previously explored. We introduce three reversing symmetries: one valid for arbitrary masses, and two that require two equal masses. We also provide a thorough characterization of their fixed points, which are crucial for understanding the dynamics of the system due to their connection with the symmetric periodic orbits of the system. Using two reversing symmetries, we numerically compute a choreography in the three-body problem on the sphere, a particular type of symmetric periodic orbit. This orbit is closely related to the classical figure-eight choreography, a well-known symmetric periodic orbit in the Newtonian planar three-body problem.