Abstract
In this study a possible application of time-dependent canonical perturbation theory to a fast nonlinear time-periodic Hamiltonian with strong internal excitation is considered. It is shown that if the time-periodic unperturbed part is quadratic, the Hamiltonian may be canonically transformed to an equivalent form in which the new unperturbed part is time-invariant so that the time-dependent canonical perturbation theory may be successfully applied. For this purpose, the Liapunov-Floquet (L-F) transformation and its inverse associated with the unperturbed time-periodic quadratic Hamiltonian are computed using a recently developed technique. Action-angle variables and time-dependent canonical perturbation theory are then utilized to find the solution in the original coordinates. The results are compared for accuracy with solutions obtained by both numerical integration and by the classical method of directly applying the time-dependent perturbation theory in which the time-periodic quadratic part is treated as another perturbation term. A strongly excited Mathieu-Hill quadratic Hamiltonian with a cubic perturbation and a nonlinear time-periodic Hamiltonian without a constant quadratic part serve as illustrative examples. It is shown that, unlike the classical method in which the internal excitation must be weak, the proposed formulation provides accurate solutions for an arbitrarily large internal excitation.
Quantum mechanical uncertainties and exact transition amplitudes for time dependent quadratic Hamiltonian