Action-angle variables are also important in obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. They only exist, providing a key characterization of the dynamics, when the system is completely integrable, i.e.,  the number of independent Poisson commuting invariants is maximal and the conserved energy surface is compact.
Action-angle variables play a vital role in calculating the frequencies of oscillatory or rotational motion without needing to solve the equations of motion. They only exist when the system is completely integrable, meaning that the number of independent Poisson commuting invariants is at its highest and the conserved energy surface is compact.