[Dynamique de l’hamiltonien dominant]
It is well known that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is defined by a basis (a collection of integer vectors). We introduce a class of resonances whose basis can be divided into two well separated groups and call them dominant. We prove that the associated slow system can be well approximated by a subsystem given by one of the groups, both in the sense of the vector field and weak KAM theory. As a corollary, we obtain perturbation results on normally hyperbolic invariant cylinders, and the Aubry/Mañe sets. This has applications in Arnold diffusion in arbitrary degrees of freedom.
Il est bien connu que les instabilités des systèmes hamiltoniens presque intégrables interviennent au voisinage des résonances. La dynamique de ces systèmes près des résonances est bien approchée par les systèmes moyennés associés, appelés systèmes lents. Chaque résonance est définie par une base (une collection de vecteurs entiers). Nous introduisons une classe de résonances dont la base peut être divisée en deux groupes bien distincts, que nous appelons dominantes. Nous prouvons que le système lent associé peut être bien approché par un sous-système donné par l’un de ces deux groupes, à la fois comme champ de vecteurs et au sens de la théorie KAM faible. Comme corollaire, nous obtenons des résultats perturbatifs sur des cylindres invariants normalement hyperboliques, et sur les ensembles d’Aubry/Mañé. Cela a des applications en diffusion d’Arnold pour un nombre arbitraire de degrés de liberté.
Keywords: Hamiltonian systems, resonant averaging, Mather theory, weak KAM theory, Arnold diffusion.
Mots-clés : Systèmes hamiltoniens, moyenne de résonance, théorie d’Mather, théorie KAM faible, diffusion d’Arnold.
Kaloshin, Vadim 1 ; Zhang, Ke 2
@article{BSMF_2018__146_3_517_0,
author = {Kaloshin, Vadim and Zhang, Ke},
title = {Dynamics of the dominant {Hamiltonian}},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {517--574},
year = {2018},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {146},
number = {3},
doi = {10.24033/bsmf.2765},
mrnumber = {3936533},
zbl = {1417.37214},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2765/}
}
TY - JOUR AU - Kaloshin, Vadim AU - Zhang, Ke TI - Dynamics of the dominant Hamiltonian JO - Bulletin de la Société Mathématique de France PY - 2018 SP - 517 EP - 574 VL - 146 IS - 3 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2765/ DO - 10.24033/bsmf.2765 LA - en ID - BSMF_2018__146_3_517_0 ER -
%0 Journal Article %A Kaloshin, Vadim %A Zhang, Ke %T Dynamics of the dominant Hamiltonian %J Bulletin de la Société Mathématique de France %D 2018 %P 517-574 %V 146 %N 3 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2765/ %R 10.24033/bsmf.2765 %G en %F BSMF_2018__146_3_517_0
Kaloshin, Vadim; Zhang, Ke. Dynamics of the dominant Hamiltonian. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 3, pp. 517-574. doi: 10.24033/bsmf.2765
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