Frequency extraction from invariant flows

In nondegenerate integrable Hamiltonian systems, invariant tori can be parametrized by action variables or fundamental frequencies. We introduce an invariant-flow formulation, consistent with the Liouville-Arnold geometry, for extracting these frequencies. Treating invariants as generators of commuting Hamiltonian flows yields the frequencies from time-of-flight parameters, providing a direct alternative to action-angle constructions and long-time spectral analysis. The resulting numerical procedure extends naturally to multiple degrees of freedom and achieves high accuracy for the McMillan map. More broadly, the framework provides an invariant-centered view of near-integrable dynamics, recovering frequencies and other dynamical information from high-quality approximate invariants.