Hamiltonian stability and subanalytic geometry

Niederman, Laurent

Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 795-813.

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.

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.

Classification : 14P15, 32B20, 32S05, 37J40, 70H08, 70H09, 70H14
Keywords: Hamiltonian systems, stability, subanalytic geometry, curve selection lemma, Lojasiewicz’s inequalities
Mot clés : systèmes Hamiltoniens, stabilité, géométrie sous-analytique, lemme du Petit Chemin, inégalités de Lojasiewicz

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