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An introduction to Nekhoroshev theory

Course Type: 
PhD Course
Academic Year: 
2023-2024
Period: 
January - March
Duration: 
20 h
Description: 
The present schedule may still be subject to little variations, mainly to minimize possible overlaps with other courses.
 
Summary:
 
When a finite dimensional Hamiltonian systems is integrable, its phase space admits a foliation in invariant tori, along which the flow evolves linearly. Are these tori stable under small size perturbations of the Hamiltonian? The celebrated KAM Theorem answers positively to this question for non resonant tori, giving stability for all times. On the other hand Nekhoroshev Theorem proves that, for exponentially long times, all tori are stable.
 
This course will present the exponential stability estimates of Nekhoroshev theorem. Two main approaches are known to its proof, enlightening different features of the dynamics: the one originally formulated by Nekhoroshev and the one due to Lochak.
The two proofs will be discussed, with particular attention to the different role played by the geometric properties (namely, steepness, or convexity/quasi-convexity) assumed on the unperturbed Hamiltonian.
In the last part of the course, extensions of Nekhoroshev theory to infinite dimensional systems will be discussed.
 
 
Location: 
Room 134 (April 23, May 3, May 10, May 13, May 17, May 20), Room 133 (April 30, May 7, May 24), Room 136 (April 16)
Next Lectures: 
Monday, May 20, 2024 - 14:00 to 16:00
Friday, May 24, 2024 - 11:00 to 13:00

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