Hamiltonian stability and subanalytic geometry

Laurent Niederman[1]

  • [1] Université Paris XI Topologie et Dynamique UMR 8628 du CNRS Bât. 425, 91405 Orsay Cedex (France) IMCCE Astronomie et Systèmes Dynamiques UMR 8028 du CNRS 77 avenue Denfert-Rochereau, 75014 Paris (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 3, page 795-813
  • ISSN: 0373-0956

Abstract

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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 h which is real analytic around a compact set in ℝ n is steep if and only if its restriction to any affine subspace of ℝ n 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.

How to cite

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Niederman, Laurent. "Hamiltonian stability and subanalytic geometry." Annales de l’institut Fourier 56.3 (2006): 795-813. <http://eudml.org/doc/10164>.

@article{Niederman2006,
abstract = {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 $h$ which is real analytic around a compact set in $\mathbb\{R\}^n$ is steep if and only if its restriction to any affine subspace of $\mathbb\{R\}^n$ 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.},
affiliation = {Université Paris XI Topologie et Dynamique UMR 8628 du CNRS Bât. 425, 91405 Orsay Cedex (France) IMCCE Astronomie et Systèmes Dynamiques UMR 8028 du CNRS 77 avenue Denfert-Rochereau, 75014 Paris (France)},
author = {Niederman, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Hamiltonian systems; stability; subanalytic geometry; curve selection lemma; Lojasiewicz’s inequalities; Lojasiewicz's inequalities},
language = {eng},
number = {3},
pages = {795-813},
publisher = {Association des Annales de l’institut Fourier},
title = {Hamiltonian stability and subanalytic geometry},
url = {http://eudml.org/doc/10164},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Niederman, Laurent
TI - Hamiltonian stability and subanalytic geometry
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 795
EP - 813
AB - 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 $h$ which is real analytic around a compact set in $\mathbb{R}^n$ is steep if and only if its restriction to any affine subspace of $\mathbb{R}^n$ 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 - eng
KW - Hamiltonian systems; stability; subanalytic geometry; curve selection lemma; Lojasiewicz’s inequalities; Lojasiewicz's inequalities
UR - http://eudml.org/doc/10164
ER -

References

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