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Abstract
Perturbative methods for constructing action-angle transformations for classical dynamical systems are examined. A new Lie perturbation theory is developed; this theory has a structure which allows its application to high order. The single sinusoidal electrostatic wave potential and the Chirikov-Taylor mapping are used as examples. It is found that the perturbation series contain resonant denominators which cause non-uniform convergence. Several renormalization techniques are attempted for the single-wave case in an effort to render the theory uniformly convergent: coordinate straining, ad hoc resonance broadening and integral representations. Each of these methods presents serious drawbacks. The reasons for non-uniform convergence of the perturbation theories are traced to singularities of the exact solution for the single wave in the complex domain of the action-angle variables. The analytic continuation of the dynamics permits the development of new pictures which may allow for the construction of uniformly convergent perturbation theories. The use of non-canonically-conjugate variables is explored in constructing perturbation theories. In particular, the angular frequency is used instead of the action as an independent variable. The Chirikov-Taylor mapping is examined with these perturbation theories to gain insight on the stochastic transition. It is argued that the Lie operator method can also be justified in this case by analytically continuing from the complex angular frequency plane onto the real line. By examining on the order of 10('2) terms it is determined that the perturbation series appears to converge on KAM surfaces. The method is not superconvergent, but yields simple recursion relations which allow automatic algebraic manipulation techniques to be used to develop the series to high order. The resulting picture is one where preserved primary KAM surfaces are continuously connected to one another in the complex angular frequency domain.
