Hamiltonian Perturbation Theory on a Manifold

Details Hamiltonian Perturbation Theory on a Manifold Hamiltonian Perturbation Theory on a Manifold

May 1995

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995cemda..62...43f&link_type=abstract

Celestial Mechanics & Dynamical Astronomy, Volume 62, Issue 1, pp.43-69

Astronomy and Astrophysics

Astronomy

4

Hamiltonian Perturbation Theory, Degenerate Hamiltonian Systems, Bifibrations, Global Action-Angle Coordinates, Nekhoroshev'S Theorem

Scientific paper

This paper deals with Hamiltonian perturbation theory for systems which, like Euler-Poinsot (the rigid body with a fixed point and no torques), are degenerate and do not possess a global system of action-angle coordinates. It turns out that the usual methods of perturbation theory, which are essentially ‘local’ being based on the construction of normal forms within the domain of a local coordinate system, are not immediately usable to study perturbations of these systems, since degeneracy makes impossible to control that the system does not fall into a singularity of the coordinates. To overcome this difficulty, we develop a ‘global’ formulation of Hamiltonian perturbation theory, in which the normal forms are globally defined on the phase space manifold. The key for this study lies in the geometry of the fibration by the invariant tori of an integrable degenerate Hamiltonian system, which is described by some generalizations of the Liouville-Arnol'd theorem and is reviewed in the paper. As an application, we provide a ‘global’ formulation of Nekhoroshev's theorem on the stability for exponentially long times.

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