Super-convergent adiabatic invariants with resonant denominators by Lie transforms

Adiabatic invariants of motion for perturbed Hamitonian systems are very important in plasma physics. It has been shown how to deal with resonant denominators to first order in the perturbation by judicious choice of the zeroth order invariant. The method is extended to higher orders by solving the Liouville equation and by using the Lie transform technique. The resulting invariants are shown to be equivalent by using the operator algebra of the averaging method involving the Poisson bracket, integrating, and averaging operations. Partial Lie transforms are introduced to describe the internal structure of any resonance in any order. The super convergent expansions of Kolmogorov are easily illustrated in the Lie formalism and allow a quick analysis of high-order resonances to be made.