Notes on exponential stability of Hamiltonian
by Antonio Giorgilli
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
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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