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.

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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DOI : 10.5802/afst.1723
Barbieri, Santiago 1

1 Institut de Mathématiques d’Orsay - Université Paris Saclay, Bât. 307, Rue Michel Magat, 91400 Orsay, France — Dipartimento di Matematica e Fisica - Università degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, Palazzina C, 00146 Roma, Italy — Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
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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/

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