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Notes on Hamiltonian Dynamical Systems
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Notes on Hamiltonian Dynamical Systems

Notes on Hamiltonian Dynamical Systems

Antonio Giorgilli

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460 pages, parution le 04/05/2022
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Résumé

Starting with the basics of Hamiltonian dynamics, this text follows the historical development of the theory culminating in key recent results: Kolmogorov's theorem, Nekhoroshev's theorem and superexponential stability. Students will find a plain introduction to perturbation methods, allowing them to learn the advanced material.Starting with the basics of Hamiltonian dynamics and canonical transformations, this text follows the historical development of the theory culminating in recent results: the Kolmogorov-Arnold-Moser theorem, Nekhoroshev's theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouville's theorem, the proof of Poincare 's non-integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincare and its relations with integrability, also including recent results on superexponential stability. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students.1. Hamiltonian formalism; 2. Canonical transformations; 3. Integrable systems; 4. First integrals; 5. Nonlinear oscillations; 6. The method of Lie series and of Lie transform; 7. The normal form of Poincare and Birkhoff; 8. Persistence of invariant tori; 9. Long time stability; 10. Stability and chaos; A. The geometry of resonances; B. A quick introduction to symplectic geometry; References; Index.Antonio Giorgilli is a retired Professor of Mathematics at the Universita degli Studi di Milano and has been elected corresponding member of Istituto Lombardo Accademia di Scienze e Lettere since 2005. He has taught courses in mathematical physics and dynamical systems ranging from undergraduate to PhD level. His research in dynamical systems focuses on the characterization of chaos, KAM theory and Nekhoroshev's theory on exponential stability. In addition to being an Invited Speaker of the 1998 International Congress of Mathematicians, Giorgilli's other notable honors include the International Gili Agostinelli Prize for Pure or Applied Mechanics or Classical Mathematical Physics by the Accademia delle Scienze di Torino in 2007. The minor planet 27855 Giorgilli, discovered in 1995, is named in his honor.

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Éditeur(s) Cambridge University Press
Auteur(s) Antonio Giorgilli
Parution 04/05/2022
Nb. de pages 460
EAN13 9781009151139

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