The Collinear Three-Body Problem with Negative Energy

The geometry of the global phase space of the collinear three-body problem with negative energy is presented in this paper. A set of transformations is introduced to create fictitious boundaries to make the phase space compact. At first, the binary collisions are not regularized. Then one of the binary collisions (the collision between m2 and m3) is regularized and we analyze the phase structure of this "half regularized" system. Finally, the second binary collision (the collision between m1 and m2) is regularized and we analyze how the phase structure is transformed by this regularization. The whole analysis provides a vivid picture of the phase flow of the collinear three-body problem.