Schirinzi, Gabriella (2014) Investigation of new conditions for steepness from a former result by Nekhoroshev. [Ph.D. thesis]

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Abstract (english)

This Thesis presents the construction of new sufficient conditions for the verification of a property of functions called steepness. It is a peculiar property required for the application of the Nekhoroshev Theorem to a quasi-integrable Hamiltonian system, and its formulation is given by Nekhoroshev in an implicit way. Therefore sufficient conditions are necessary for the verification of the steepness.
Nekhoroshev formulated his celebrated Theorem in the seventies, providing under suitable hypothesis a strong stability result for those dynamical systems which are not integrable, but can be considered as a small perturbation of an integrable system. The Nekhoroshev Theorem is a fundamental result in the framework of the Perturbation Theory, especially for its important applications in Celestial Mechanics.
For the construction of new sufficient conditions for steepness, a result proved by Nekhoroshev is used. The new conditions are weaker than the ones known up to now, hence they allow to detect a larger class of steep functions. In particular, the new conditions concern functions of two, three and four variables respectively.
In the last Chapter of this Thesis a general algorithm for the verification of the steepness of functions of three or four variables is constructed. Moreover, in order to provide some concrete examples of applicability of the new conditions, such algorithm is applied to two physical systems: the Hamiltonian of the circular restricted three-body problem, and the Hamiltonian of a chain of four harmonic oscillators, with the potential energy of the Fermi-Pasta-Ulam problem. In both cases the new sufficient conditions allow to prove numerical evidence of the steepness.

Abstract (italian)

In questa tesi viene presentata la costruzione di nuove condizioni sufficienti per la verifica di una proprietà delle funzioni denominata steepness. Tale proprietà è un’ipotesi fondamentale per l’applicazione del teorema di Nekhoroshev ad un sistema Hamiltoniano quasi integrabile, e la sua formulazione viene fornita da Nekhoroshev in maniera implicita. Per questo motivo è necessario avere a disposizione delle condizioni sufficienti per la verifica della stepness.
Nekhoroshev formulò negli anni settanta il suo celebre teorema, il quale garantisce sotto opportune ipotesi una forte stabilità per quei sistemi dinamici che non sono integrabili, ma possono scriversi come una piccola perturbazione di un sistema integrabile. Il teorema di Nekhoroshev costituisce un risultato fondamentale nell’ambito della Teoria delle Perturbazioni, in particolar modo per le sue importanti applicazioni nella meccanica celeste.
Per la costruzione delle nuove condizioni sufficienti per la steepness viene utilizzato un risultato dimostrato da Nekhoroshev. Le nuove condizioni sono più deboli di quelle conosciute fino ad ora, e di conseguenza permettono di individuare una classe più ampia di funzioni steep. In particolare, le nuove condizioni riguardano funzioni di due, tre e quattro variabili rispettivamente.
Nell’ultimo capitolo di questa tesi viene costruito un algoritmo generale per la verifica della steepness di funzioni di tre o quattro variabili. Inoltre, allo scopo di fornire qualche esempio concreto di applicazione delle nuove condizioni, tale algoritmo viene applicato a due sistemi fisici: l’Hamiltoniana del problema dei tre corpi ristretto circolare, e l’Hamiltoniana di una catena di quattro oscillatori armonici, con l’energia potenziale del problema di Fermi-Pasta-Ulam. In entrambi i casi le nuove condizioni sufficienti permettono di dimostrare numericamente
la steepness.

EPrint type: Ph.D. thesis Guzzo, Massimiliano Ciclo 26 > Scuole 26 > SCIENZE MATEMATICHE > MATEMATICA COMPUTAZIONALE 24 January 2014 24 January 2014 Hamiltonian systems, Nekhoroshev Theorem, Perturbation Theory, Steepness, stability, celestial mechanics, quasi-integrable systems Area 01 - Scienze matematiche e informatiche > MAT/07 Fisica matematica Dipartimenti > Dipartimento di Matematica 6350 04 Nov 2014 13:54
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