# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Persistence of invariant torus in Hamiltonian systems with two-degree of freedom. (English) Zbl 1134.37025
Consider the following Hamiltonian dynamical system: $·q={H}_{p}\left(p,q\right),\phantom{\rule{1.em}{0ex}}·p=-{H}_{q}\left(p,q\right)$ where the Hamiltonian function is $H=h\left(p\right)+f\left(q,p\right)$. The classical KAM theorem asserts that if $h$ is not degenerate i.e. det$\left({h}_{pp}\right)\ne 0$ 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.
##### MSC:
 37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion 70H08 Nearly integrable Hamiltonian systems, KAM theory
##### Keywords:
KAM theorem; KAM iteration