A Global Kam-Theorem: Monodromy in Near-Integrable Perturbations of Spherical Pendulum

Henk W. Broer (Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen)

Journal of Mathematical and Fundamental Sciences
Vol. 34 No. 2&3 (2002)

The KAM Theory for the persistence of Lagrangean invariant tori in nearly integrable Hamiltonian systems is lobalized to bundles of invariant tori. This leads to globally well-defined conjugations between near-integrable systems and their integrable approximations, defined on nowhere dense sets of positive measure associated to Diophantine frequency vectors. These conjugations are Whitney smooth diffeomorphisms between the corresponding torus bundles. Thus the geometry of the integrable torus bundle is inherited by the near-integrable perturbation. This is of intereet in cases where these bundles are nontrivial. The paper deals with the spherical pendulum as a leading example.