, Volume 1, Issue 3, pp 351-366

Kam-stable Hamiltonians

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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.