The symmetric elliptic and hyperbolic restricted 3-body problem on the unit circle

We study the restricted 3-body problem with the constriction of motion to the unit circle. First, we study the 2-body problem on the unit circle and give the explicit solutions for a regularized version of the equations of motion for any initial data. We classify the motions in elliptic, parabolic, hyperbolic type and an equilibrium state. Then, we analyze the restricted 3-body problem on the unit circle when the primary bodies are performing elliptic and hyperbolic motions. We show the existence of just one equilibrium state when the masses of primary bodies are equal and we exhibit the hyperbolic structure of this equilibrium point via an exponential dichotomy. In the last part we regularize the equations of motion. We show the global dynamics and some periodic solutions with its respective period.