Non-quadratic additional conserved quantities in Birkhoff normal forms

Author: Gaeta, Giuseppe

Source: Celestial Mechanics and Dynamical Astronomy, Volume 96, Number 1, September 2006 , pp. 63-81(19)

Publisher: Springer

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Abstract:

For resonant Hamiltonian systems in Poincaré-Birkhoff normal form, the quadratic part of the Hamiltonian is a constant of motion. In the resonant case, the normal form is not unique; this corresponds to free parameters in the solution to homological equations. The “standard” prescription in this case is to set these parameters to zero; however, it was remarked already by Dulac that a different prescription could actually produce a simpler normal form. One such prescription was provided in previous work by the present author; here we discuss how—and under which conditions—this can be used to obtain normal forms which admit, besides the quadratic part, (one or a set of) additional constants of motion of higher degree in nested small neighborhoods of the origin. A concrete example with a cubic natural Hamiltonian in 3 DOF is considered.

Keywords: Hamiltonian systems; Constants of motion; Perturbation theory; Birkhoff normal forms

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10569-006-9026-9

Affiliations: 1: Email: gaeta@mat.unimi.it

Publication date: 2006-09-01