This Demonstration shows the phase portrait of the Hamilton equations that describe the motion of a particle of mass and charge in the field of a standing wave. You can see separatrices corresponding to resonance with waves.

Choosing and defining gives a time-dependent Hamiltonian

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Reference

[1] J. Candy and R. Rozmus, "A Symplectic Integration Algorithm for Separable Hamiltonian Functions," Journal of Computational Physics, 92(1), 1991 pp. 230–256. doi:10.1016/0021-9991(91)90299-Z.