Successive Elimination of Harmonics: A way to Explore the Resonant Structure of a Hamiltonian System
This paper deals with the problem of exploring Hamiltonian systems with semi-numerical methods. I come back to an original idea of Delaunay, consisting in successively eliminating all the perturbation harmonics up to a given order. Here the elimination is performed via the introduction of suitable Arnold action-angle variables, along the lines of the well known theorem of Arnold (1963b), with numerical evaluation of the action integrals. More precisely, given the Fourier expansion of the Hamiltonian, I first ignore all the harmonics in the perturbation except a single one; this is an inte-grable system and the introduction of Arnold action-angle variables allows the complete elimination of the considered harmonic without remainder terms. Next, I compute the Fourier expansion of the till now ignored terms in the new variables. This algorithm can be implemented on computer and iterated so that one attains a global description of the dynamics. By the way, this allows to point out all the power of the method of Delaunay, which seems to have been almost forgotten.
KeywordsFourier Expansion Invariant Torus Main Resonance Asteroid Belt Successive Elimination
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