  Integration of Hamiltonian systems by reduction to action-angle variables has
proven to be a successful approach. However, when the solution depends on
elliptic functions the transformation to action-angle variables may need to
remain in implicit form. This is exactly the case of the simple pendulum, where
in order to make explicit the transformation to action-angle variables one
needs to resort to nontrivial expansions of special functions and series
reversion. Alternatively, it is shown that the explicit expansion of the
transformation to action-angle variables can be constructed directly, and that
this direct construction leads naturally to the Lie transforms method, in this
way avoiding the intricacies related to the traditional expansion of elliptic
functions.
