Journal of Dynamical and Control Systems

, Volume 1, Issue 3, pp 351–366

Kam-stable Hamiltonians

Authors

  • M. B. Sevryuk
    • Institite of Energy Problems of Chemical Physics
Article

DOI: 10.1007/BF02269374

Cite this article as:
Sevryuk, M.B. Journal of Dynamical and Control Systems (1995) 1: 351. doi:10.1007/BF02269374

Abstract

We present a simple proof of Rüssmann's theorem on invariant tori of analytic perturbations of analytic integrable Hamiltonian systems of the formdp/dt=0,dq/dtf(p)p, where (p, q) are the action-angle variables. Rüssmann's theorem asserts that if the image of the mappingp→ϖf(p)p does not lie in any linear hyperplane passing through the origin, then any sufficiently small Hamiltonian perturbation of this integrable system possesses many invariant tori close to the unperturbed tori {p=const}. The main idea of our proof is that we embed the perturbed Hamiltomian in a family of Hamiltonians depending on an external multidimensional parameter. We also show that the Rüssmann condition is necessary (i.e., not only sufficient) for the existence of perturbed tori and give analogs of Rüssmann's theorem for exact symplectic diffeomorphisms, reversible flows, and reversible diffeomorphisms.

1991 Mathematics Subject Classification

58F2758F3058F0534C50

Key words and phrases

Nearly integrable Hamiltonian systemsinvariant torifrequency mapKAM-stability

Copyright information

© Plenum Publishing Corporation 1995