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://www.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://www.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://www.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://www.numdam.org/articles/10.5802/afst.1723/
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