Action-angle variables for geodesic motions in Sasaki-Einstein spaces $Y^{p,q}$

We use action-angle variables to describe the geodesic motions in the 5-dimensional Sasaki-Einstein spaces $Y^{p,q}$. This formulation allows us to study thoroughly the complete integrability of the system. We find that the Hamiltonian involves a reduced number of action variables. Therefore one of the fundamental frequencies is zero, indicating chaotic behavior when the system is perturbed.