The classical averaging theories founded on Jacobian canonical
transformations
(viz. Lindstedt, Von Zeipel, Delaunay) cannot be extended to allow the
study of
the neighbourhood of a resonance, except in the case of only one degree of
freedom
(e.g. Garfinkel's Ideal Resonance Problem). When long-period variables are
also
considered leading to increase the number of degrees of freedom, one
singularity
appear on the right-hand side of the perturbation equations, as shown by
Poincaré.
Some of the attempts to circumvent this problem in the past are discussed
and
compared to modern methods founded on Lie series.