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Persistence of invariant torus in Hamiltonian systems with two-degree of freedom. (English) Zbl 1134.37025
Consider the following Hamiltonian dynamical system: $.{q}=H_p(p,q), \quad .{p}=-H_q(p,q)$ where the Hamiltonian function is $H=h(p)+f(q,p)$. The classical KAM theorem asserts that if $h$ is not degenerate i.e. det$(h_{pp})\neq 0$ then most of the invariant tori can persist when $f$ is sufficiently small. In general, the nondegeneracy condition is necessary for KAM theorems. However, the Hamiltonian systems with two degrees of freedom have some special properties and so, the present paper is devoted to a KAM theorem for a class of 2D Hamiltonian systems without any nondegeneracy condition. The main tool is the so-called KAM iteration.

37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H08Nearly integrable Hamiltonian systems, KAM theory
Full Text: DOI
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