National Institute of Standards and Technology, Gaithersburg, MD 20899-1000, USA
The fundamental operation of Poincare's methode nouvelle is the expansion in powers of a small parameter \epsilon for a power series in \epsilon that is a function whose variables are themselves series in \epsilon. An inductive scheme is set up for that type of expansions. The scheme is applied to build the partial differential equations by which Poincare intended to determine beyond order one the generator of the transformation as well as the transformed Hamiltonian.
The scheme renders manifest the advantage of the methode nouvelle over a treatment of perturbations by Lie transformations. In both methods, partial differentiation with respect to the coordinates act on the original Hamiltonian; In Poincares algorithm, however, differentiation with respect to the moments bear on the transformed Hamiltonian -- usually simpler in form than the original one. This is, of course, not the case for Lie transformations.
Disaffection among astronomers towards Poincare's procedure came from the fact that an explicit representation of the transformation and its inverse requires inversion of series and substitutions of series into series. Recent advances in Symbolic Algebra have brought about a systematic solution to these problems in the most general frame of n >= 1 implicit equations in n unknowns dependent on m >= 1 variables.
The methode nouvelle has been given a new chance in life.