Un système hamiltonien presque intégrable est stable sur un temps exponentiellement long s’il est holomorphe et si sa partie intégrable satisfait à une propriété géométrique générique dite d’escarpement (steepness). Suivant une stratégie développée par Nekhoroshev, on donne des conditions algébriques suffisantes pour garantir qu’une fonction donnée est escarpée, ce qui est important en vue des applications, notamment en mécanique céleste. Ces conditions portent sur les dérivées jusqu’à l’ordre cinq de la fonction étudiée. L’étude de la théorie sous-jacente permet des considérations intéressantes sur la généricité de la propriété d’escarpement. De plus, ce travail représente un premier pas vers la construction de conditions qui garantissent l’escarpement d’une fonction donnée et qui portent sur ses dérivées à un ordre arbitraire.
Steepness is a generic geometric property which, together with complex-analyticity, is needed in order to ensure stability of a nearly-integrable hamiltonian system over exponentially long times. Following a strategy developed by Nekhoroshev, we construct sufficient conditions for steepness of a given function that involve algebraic equations on its derivatives up to order five. This is important in view of applications (e.g. in Celestial Mechanics). The underlying analysis suggests some interesting considerations on the genericity of steepness. Moreover, this work represents a first step towards the construction of sufficient conditions for steepness involving the derivatives of the studied function up to an arbitrary order.
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@article{AFST_2022_6_31_5_1365_0,
author = {Barbieri, Santiago},
title = {On the algebraic properties of exponentially stable integrable hamiltonian systems},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1365--1390},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 31},
number = {5},
year = {2022},
doi = {10.5802/afst.1723},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/afst.1723/}
}
TY - JOUR AU - Barbieri, Santiago TI - On the algebraic properties of exponentially stable integrable hamiltonian systems JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2022 SP - 1365 EP - 1390 VL - 31 IS - 5 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1723/ DO - 10.5802/afst.1723 LA - en ID - AFST_2022_6_31_5_1365_0 ER -
%0 Journal Article %A Barbieri, Santiago %T On the algebraic properties of exponentially stable integrable hamiltonian systems %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2022 %P 1365-1390 %V 31 %N 5 %I Université Paul Sabatier, Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1723/ %R 10.5802/afst.1723 %G en %F AFST_2022_6_31_5_1365_0
Barbieri, Santiago. On the algebraic properties of exponentially stable integrable hamiltonian systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 5, pp. 1365-1390. doi : 10.5802/afst.1723. http://archive.numdam.org/articles/10.5802/afst.1723/
[1] Instability of dynamical systems with several degrees of freedom, Sov. Math., Dokl. (1964) no. 5, pp. 581-585 | Zbl
[2] Mathematical Aspects of Classical and Celestial Mechanics, Springer, 2010
[3] Superexponential Stability of Quasi-Periodic Motion in Hamiltonian Systems, Commun. Math. Phys., Volume 350 (2017) no. 1, pp. 361-386 | DOI | MR | Zbl
[4] Generic fast diffusion for a class of non-convex hamiltonians with two degrees of freedom, Mosc. Math. J., Volume 14 (2014) no. 2, pp. 181-203 | DOI | MR | Zbl
[5] Kolmogorov’s 1954 paper on nearly-integrable Hamiltonian systems, Regul. Chaotic Dyn., Volume 13 (2008), pp. 130-139 | DOI | Zbl
[6] On steepness of 3-jet non-degenerate functions, Ann. Mat. Pura Appl., Volume 198 (2019) no. 6, pp. 2151-2165 | DOI | MR | Zbl
[7] Construction of a Nekhoroshev like result for the asteroid belt dynamical system, Celest. Mech. Dyn. Astron., Volume 66 (1997) no. 3, pp. 255-292 | DOI | MR | Zbl
[8] Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems, Publ. Math., Inst. Hautes Étud. Sci. (2003) no. 96, pp. 199-275 | DOI | Numdam | MR | Zbl
[9] Stable lower estimates for smooth mapping and for gradients of smooth functions, Math. USSR, Sb., Volume 19 (1973) no. 3, pp. 425-467
[10] An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. I, Russ. Math. Surv., Volume 32 (1977) no. 6, pp. 1-65 | DOI | MR | Zbl
[11] An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. II, Topics in Modern Mathematics, Petrovskii Seminar (1979) no. 5, pp. 5-50 | MR
[12] Hamiltonian stability and subanalytic geometry, Ann. Inst. Fourier, Volume 56 (2006) no. 3, pp. 795-813 | DOI | Numdam | MR | Zbl
[13] Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dyn. Syst., Volume 27 (2007) no. 3, pp. 905-928 | DOI | MR | Zbl
[14] Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., Volume 18 (2013) no. 6, pp. 860-906 | DOI | MR | Zbl
[15] On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev, J. Math. Phys., Volume 54 (2013) no. 7, 072702, 23 pages | MR | Zbl
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