[Stabilité halmiltonienne et géométrie subanalytique]
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In the 70’s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Using theorems of real subanalytic geometry, we derive a geometric criterion for steepness: a numerical function which is real analytic around a compact set in is steep if and only if its restriction to any affine subspace of admits only isolated critical points. We also state a necessary condition for exponential stability, which is close to steepness.
Finally, we give methods to compute lower bounds for the steepness indices of an arbitrary steep function.
La notion de raideur a été introduite pour étudier la stabilité effective des systèmes Hamiltoniens quasi-intégrables. À l’aide de théorèmes de géométrie sous-analytique, on donne une condition géométrique simple qui est équivalente à la raideur pour une fonction réelle analytique.
Keywords: Hamiltonian systems, stability, subanalytic geometry, curve selection lemma, Lojasiewicz’s inequalities
Mots-clés : systèmes Hamiltoniens, stabilité, géométrie sous-analytique, lemme du Petit Chemin, inégalités de Lojasiewicz
Niederman, Laurent 1
@article{AIF_2006__56_3_795_0,
author = {Niederman, Laurent},
title = {Hamiltonian stability and subanalytic geometry},
journal = {Annales de l'Institut Fourier},
pages = {795--813},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {56},
number = {3},
year = {2006},
doi = {10.5802/aif.2200},
zbl = {1120.14048},
mrnumber = {2244230},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/aif.2200/}
}
TY - JOUR AU - Niederman, Laurent TI - Hamiltonian stability and subanalytic geometry JO - Annales de l'Institut Fourier PY - 2006 SP - 795 EP - 813 VL - 56 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://geodesic.mathdoc.fr/articles/10.5802/aif.2200/ DO - 10.5802/aif.2200 LA - en ID - AIF_2006__56_3_795_0 ER -
%0 Journal Article %A Niederman, Laurent %T Hamiltonian stability and subanalytic geometry %J Annales de l'Institut Fourier %D 2006 %P 795-813 %V 56 %N 3 %I Association des Annales de l’institut Fourier %U http://geodesic.mathdoc.fr/articles/10.5802/aif.2200/ %R 10.5802/aif.2200 %G en %F AIF_2006__56_3_795_0
Niederman, Laurent. Hamiltonian stability and subanalytic geometry. Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 795-813. doi: 10.5802/aif.2200
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