Poincaré's méthode nouvelle by skew composition

Details Poincaré's méthode nouvelle by skew composition Poincaré's méthode nouvelle by skew composition

Jul 1999

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999cemda..74..175d&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, v. 74, Issue 3, p. 175-197 (1999).

Astronomy and Astrophysics

Astronomy

Hamiltonian Dynamics, Perturbation Theory, Implicit Equations, Lagrange'S Formula, Symbolic Algebra, Hamiltonian Dynamics, Perturbation Theory, Implicit Equations, Lagrange'S Formula, Symbolic Algebra

Scientific paper

Poincaré designed the méthode nouvelle in order to build approximate integrals of Hamiltonians developed as series of a small parameter. Due to several critical deficiencies, however, the method has fallen into disuse in favor of techniques based on Lie transformations. The paper shows how to repair these shortcomings in order to give Poincaré’s méthode nouvelle the same functionality as the Lie transformations. This is done notably with two new operations over power series: a skew composition to expand series whose coefficients are themselves series, and a skew reversion to solve implicit vector equations involving power series. These operations generalize both Arbogast’s technique and Lagrange’s inversion formula to the fullest extent possible.

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