Notes on exponential stability of Hamiltonian systems
by Antonio Giorgilli

Preface: The present notes contain the full text of a series of lectures that I presented at the Centro di Ricerca Matematica Ennio De Giorgi during a Research Trimester on Dynamical Systems. The main purpose of the lectures was to present the theorem of Nekhoroshev on stability over exponentially long times. The text is organized in five chapters. Two chapters, namely the first and the fourth one, have a introductory character. I just recall some technicalities concerned with the Hamiltonian formalism in Classical Mechanics and the use of Lie Transforms in Perturbation Theory. Two chapters, the second and the third one, discuss the preliminaries to Nekhoroshev's theory, namely the theorem of Arnold-Jost on integrable systems and the theorem of Poincaré on non-existence of first integrals that may be expressed as power series in a perturbation parameter. A quantitative discussion of the simple case of an elliptic equilibrium is also included, with the aim of introducing the concept of exponential stability on a very simple framework. The last chapter contains a complete proof of Nekhoroshev's theorem along the lines of the original paper of Nekhoroshev.
Presenting this series of lectures was a great pleasure for me. With the present notes I want to express my gratitude to the organizers of the Research Trimester for having offered me that opportunity..
Finally, I want to express my appreciation for the contribution of Ugo Locatelli, who had the patience of reading the whole set of notes and of contributing in a substantial way to the improvement of the text - and to the correction of several mistakes.


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