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

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