  The merits of a perturbation theory based on a mean to osculating
transformation that is pure periodic in the fast angle are investigated. The
exact separation of the purely short-period effects of the perturbed Keplerian
dynamics from the long-period mean frequencies is achieved by a non-canonical
transformation, which, therefore, cannot be computed by Hamiltonian methods.
For this case, the evolution of the mean elements strictly adheres to the
average behavior of the osculating orbit. However, due to the inescapable
truncation of perturbation solutions, the fact that this theory confines the
long-period variations of the semimajor axis into the mean variation equations,
how tiny they may be, can have adverse effects in the accuracy of long-term
semi-analytic propagations based on it
