Classical and semiclassical mechanics of strongly resonant systems: A Fourier transform approach

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Abstract
The Fourier transform approach to EBK quantization, previously applied to nonresonant systems with up to four degrees of freedom [J. Chem. Phys. 83, 2990 (1985)], is extended to the case of strongly resonant classical motion. The classical mechanics of systems with 3:4, 1:2, and 1:1 resonances is examined in detail from the Fourier transform point of view, and the results of nonlinear resonance analysis used to interpret numerical trajectory Fourier spectra. Calculation of classical actions and numerical construction of the angle parametrization of invariant tori is described, and the relation between spectral frequency assignments and the choice of good action-angle variables investigated. It is shown that correct quantization conditions for arbitrary resonant motion can be determined by direct numerical evaluation of Maslov indices. Semiclassical eigenvalues are reported for the 3:4, 1:2, and 1:1 resonant systems.