University of Sao Paulo, Brazil
The averaging theories devised in Celestial Mechanics, since introduced by Delaunay in his celebrated La Theorie de la Lune, are generally founded on the powerful tools of Hamiltonian mechanics. In the past 30 years, the basis of Hamiltonian averaging theories has been refurbished by the introduction of Lie series mappings to represent the canonical transformations.
Lie series mappings have been cherished from the beginning for the fact that they allow canonical transformations to be written with explicit equations and provide formal schemes highly suited for automatic calculations. But they have a lot more of qualities. At variance with the former theories using Jacobian canonical mappings, theories founded on Lie series mappings are not tied to action-angle variables. They may be set with any canonical variables, (for example, Poincare's non-singular variables). For this reason, they allowed more generic developments putting into evidence important features as the existence of an integrable system with nuclear properties (the Hori's kernel). The dynamics of this auxiliary system is reproduced in the solutions built with the averaging theory, at any order, what means that these theories are only able to accurately construct solutions whose dynamics is previously fixed, what can be done by a suitable tailoring of the Hori' kernel of the theory.
Lie series mappings also allowed to decompose the perturbation equations following identity rules not founded exclusively on the power of the small parameter (disturbing mass).
These properties allowed the construction of generalized theories which give the solutions in the neighbourhood of resonances with a formal rigor not possible with former theories.