Poincaré's méthode nouvelle by skew composition
Poincaré's méthode nouvelle treats conservative Hamiltonians in the form of power series in a small parameter \(\varepsilon \), and constructs the transformed Hamiltonian and generating function also as power series in \(\varepsilon \). The lack of a systematic symbolic algorithm and some other fundamental deficiencies compelled dynamicists to abandon this method and to decide for techniques based on Lie transformations. Here the authors propose several improvements in order to give to Poincaré's method the flexibility of perturbation procedures based on Lie transformations. The improvements are mainly based on two new operations on power series (a skew composition to expand series whose coefficients are also series, and a skew inversion to solve implicit vector equations involving power series), include a procedure to express the Hamiltonian in terms of partial derivatives of generator, and use an algorithm to make explicit the direct and inverse transformations defined implicitly by the generating function. Some examples show in detail how to perform these programs.
scientific article; zbMATH DE number 1442322