The word "average" and its variations became popular in the sixties and imp
licitly carried the idea that "averaging" methods lead to "average" Hamilto
nians. However, given the Hamiltonian H = H-0(J) + epsilon R(theta, J), (ep
silon < < 1), the problem of transforming it into a new Hamiltonian H* (J*)
(dependent only on the new actions J*), through a canonical transformation
given by zero-average trigonometrical series has no general solution at or
ders higher than the first.