Abstract
The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter. Lie transforms define naturally a class of canonical mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets; the inverse transformation can be built the same way.
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Deprit, A. Canonical transformations depending on a small parameter. Celestial Mechanics 1, 12–30 (1969). https://doi.org/10.1007/BF01230629
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DOI: https://doi.org/10.1007/BF01230629