Investigation of new conditions for steepness from a former result by Nekhoroshev

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. ; 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 ...