A Version of Thirring's Approach to the KAM Theorem for Quadratic Hamiltonians with Degenerate Twist

Abstract

We give a proof of the KAM theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring's proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact, drastically simplifies Thirring's proof.

6 pages, RevTex

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Cited by in corpus (7)

• An approximate renormalization-group transformation for Hamiltonian systems with three degrees of freedom
• Critical Attractor and Universality in a Renormalization Scheme for Three Frequency Hamiltonian Systems
• Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
• Approximate renormalization for the break-up of invariant tori with three frequencies
• Strange attractor for the renormalization flow for invariant tori of Hamiltonian systems with two generic frequencies
• Scaling law for the critical function of an approximate renormalization
• KAM--renormalization-group for Hamiltonian systems with two degrees of freedom