Classical Adiabatic Perturbation Theory

Methods of classical perturbation theory developed for small perturbations are extended to slowly (or adiabatically) perturbed systems, with slow dependence either on time or on dynamical variables. Specifically, the extension is performed for the canonical perturbation theory of Poincare and Von Zeipel, for the Krylov‐Bogoliubov‐Kruskal method of eliminating angle variables, for the general form of direct near‐identity canonical transformations and for two of its realizations, based on the ``conventional'' generating function and on the Lie transform. In addition, the concepts of slow (or adiabatic) perturbations and of an implicit ``small parameter'' e are clarified, as is the distinction between two alternative definitions of adiabatic invariance, and as an example the solution of the slow perturbed harmonic oscillator up to and including O(e3) is derived.