The Schr�dinger equation and canonical perturbation theory

1987 ◽  
Vol 108 (1) ◽  
pp. 25-40 ◽  
Author(s):  
Sandro Graffi ◽  
Thierry Paul
Keyword(s):  
Perturbation Theory ◽  
Dinger Equation ◽  
Canonical Perturbation

Canonical Perturbation of a Fast Time-Periodic Hamiltonian via Liapunov-Floquet Transformation

1997 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Perturbation Theory ◽  
Classical Method ◽  
Time Dependent ◽  
Fast Time ◽  
Quadratic Part ◽  
Canonical Perturbation ◽  
Internal Excitation ◽  
Quadratic Hamiltonian ◽  
Unperturbed Part ◽  
Time Periodic

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.

Long-term evolution of Galileo operational orbits by canonical perturbation theory

2014 ◽  
Vol 94 (2) ◽  
pp. 646-655 ◽  
Author(s):  
Martin Lara ◽  
Juan F. San-Juan ◽  
Luis M. López-Ochoa ◽  
Paul Cefola
Keyword(s):  
Perturbation Theory ◽  
Long Term Evolution ◽  
Term Evolution ◽  
Canonical Perturbation
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