Non-quadratic additional conserved quantities in Birkhoff normal forms

Details Non-quadratic additional conserved quantities in Birkhoff normal forms Non-quadratic additional conserved quantities in Birkhoff normal forms

Sep 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cemda..96...63g&link_type=abstract

Celestial Mechanics and Dynamical Astronomy, Volume 96, Issue 1, pp.63-81

Astronomy and Astrophysics

Astronomy

Hamiltonian Systems, Constants Of Motion, Perturbation Theory, Birkhoff Normal Forms

Scientific paper

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.

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