Notes on Hamiltonian Dynamical Systems

✍ Scribed by Antonio Giorgilli

• Publisher: Cambridge University Press
• Year: 2022
• Tongue: English
• Leaves: 472
• Series: London Mathematical Society Student Texts 102
• Edition: 1
• Category: Library

✦ Synopsis

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

Contents
Apology
Plan of the Book
Expressions of Gratitude
1 Hamiltonian Formalism
1.1 Phase Space and Hamilton's Equations
1.2 Dynamical Variables and First Integrals
1.3 Use of First Integrals
2 Canonical Transformations
2.1 Preserving the Hamiltonian Form of the Equations
2.2 Differential Forms and Integral Invariants
2.3 Generating Functions
2.4 Time-Dependent Canonical Transformations
2.5 The Hamilton-Jacobi Equation
3 Integrable Systems
3.1 Involution Systems
3.2 Liouville's Theorem
3.3 On Manifolds with Non-Singular Vector Fields
3.4 Action-Angle Variables
3.5 The Arnold-Jost Theorem
3.6 Delaunay Variables for the Keplerian Problem
3.7 The Linear Chain
3.8 The Toda Lattice
4 First Integrals
4.1 PerioDiffc and Quasi-PerioDiffc Motion on a Torus
4.2 The Kronecker Map
4.3 ErgoDiffc Properties of the Kronecker Flow
4.4 Isochronous and Anisochronous Systems
4.5 The Theorem of Poincaré
4.6 Some Remarks on thé Theorem of Poincaré
5 Nonlinear Oscillations
5.1 Normal Form for Linear Systems
5.2 Non-linear Elliptic Equilibrium
5.3 Old-Fashioned Numerical Exploration
5.4 Quantitative Estimates
6 The Method of Lie Series and of Lie Transforms
6.1 Formal Expansions
6.2 Lie Series
6.3 Lie Transforms
6.4 Analytic Framework
6.5 Analyticity of Lie Series
6.6 Analyticity of the Lie Transforms
6.7 Weighted Fourier Norms
7 The Normal Form of Poincaré and Birkhoff
7.1 The Case of an Elliptic Equilibrium
7.2 Action-Angle Variables for the Elliptic Equilibrium
7.3 The General Problem
7.4 The Dark Side of Small Diffvisors
8 Persistence of Invariant Tori
8.1 The Work of Kolmogorov
8.2 The Proof AccorDiffng to the Scheme of Kolmogorov
8.3 A Proof in Classical Style
8.4 ConcluDiffng Remarks
9 Long Time Stability
9.1 Overview on the Concept of Stability
9.2 The Theorem of Nekhoroshev
9.3 Analytic Part
9.4 Geometric Part
9.5 The Exponential Estimates
9.6 An Alternative Proof by Lochak
10 Stability and Chaos
10.1 The Neighbourhood of an Invariant Torus
10.2 The Roots of Chaos and Diffusion
10.3 An Example in Diffmension 2: the Standard Map
10.4 Stability in the Large
10.5 Some Final Considerations
Appendix A. The Geometry of Resonances
A.1 Discrete Subgroups and Resonance Moduli
A.2 Strong Non-resonance
AppenDiffx B. A Quick Introduction to Symplectic Geometry
B.1 Basic Elements of Symplectic Geometry
References
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Index
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