Global symplectic regularization for some restricted 3-body problems on S1

We develop a regularization for binary collisions in some restricted 3-body problems moving in planar one-dimensional spaces with constant curvature. The main characteristic of the regularization is that it preserves the Hamiltonian structure of the equations and it regularizes all the binary collisions with just one transformation. We apply this global symplectic regularization to the 2-body problem on the unit circle and we show the global dynamics. Also, we tackle the restricted 3-body problem with one fixed center in the unit circle and we give the global dynamics for the case when it has two fixed centers.