Notes on Hamiltonian Dynamical Systems

Antonio Giorgilli

• Publisher: Cambridge University Press
• Year: 2022
• Print ISBN: 9781009151146
• eBook ISBN: 9781009174862
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

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 Poincaré'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 Poincaré 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.

Table of contents

• Cover
• Endorsements
• Series Page
• Title Page
• Copyright Page
• Contents
• Apology
• Plan of the Book
• Expressions of Gratitude
• 1 Hamiltonian Formalism
• 2 Canonical Transformations
• 3 Integrable Systems
• 4 First Integrals
• 5 Nonlinear Oscillations
• 6 The Method of Lie Series and of Lie Trans-forms
• 7 The Normal Form of Poincaré and Birkho
• 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