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International scientific conference "Days of Classical Mechanics"
January 26, 2015 11:25–12:00, Moscow, Steklov Mathematical Institute of RAS, Gubkina, 8

Lower dimensional tori in Hamiltonian systems. Variational approach

P. I. Plotnikov, I. V. Kuznetsov
 Video records: Flash Video 219.9 Mb Flash Video 1,316.8 Mb MP4 219.9 Mb

Abstract: In this report the problem of persistence of invariant tori of codimension $k\geq 1$ with a fixed frequency vector under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between weakly hyperbolic invariant tori and critical points of the Parceval potential defined on $\mathbb T^k\times [0,1]^k$ is established. We give sufficient conditions which being imposed on the unperturbed Hamiltonian provide the existence at least one weakly hyperbolic invariant torus. We also give an example of a convex unperturbed Hamiltonian for which all perturbed low dimensional invariant tori are elliptic.