Advanced Hamiltonian Dynamics and Arnold Diffusion by Cheng, Chong-Qing ; Xue, Jinxin;

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This book presents the advanced theory of Hamiltonian dynamics, with an emphasis on the recent development of variational methods and its application to the problem of Arnold diffusion. The main theme of the book is to study the dynamics of finite dimensional nearly integrable Hamiltonian systems, which are small perturbations of integrable systems.

The book consists of two parts. Part I includes the main techniques in Hamiltonian dynamics such as the integrability and nonintegrability theory, the normal form theory (KAM theory, Nekhoroshev theorem), the hyperbolicity theory (the theorem of normally hyperbolic invariant manifold), the variational theory, systems of two degrees of freedom and the connecting orbit theory. In the more advanced Part II, authors specialize to the proof of Arnold diffusion. The techniques in Part I are fully exploited in Part II for people to understand theorems better via applications.The proof the classical Tonelli theorem, some preliminaries of ergodic theory and some basics of genercitity and transversality are given in the Appendix.

This book can be used as the textbook for graduate students in the field of Hamiltonian dynamics. It can also be used as reference for working mathematicians. Before reading this book, obtaining some understanding on Arnold's book Mathematical Methods in Classical Mechanics, algebraic topology, differential topology and convex analysis will be helpful.

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Part I Advanced Hamiltonian Dynamics.- Chapter 1. Integrability and Nonintegrability.- 1.1 The Lagrangian, Hamiltonian and Hamilton-Jacobi PDE formalisms.- 1.2 The algebraic, symplectic and geometric structures.- 1.3 Dynamical properties: Liouville Theorem and Poincare recurrence.- 1.4 Integrable systems, the Liouville-Arnold theorem.- 1.5 Newtonian two-body problem*.- 1.6 Poincare’s theorem: onset of chaos.- 1.7 Further examples of integrable systems*.- Chapter 2. The Normal Form Theory.- 2.1 The N-body problem*.- 2.2 The Kolmogorov-Arnold-Moser (KAM) theory.- 2.3 The Nekhoroshev theorem.- 2.4 Resonance produces pendulum: the normal form package.- 2.5 Complete resonance, synchronization and closing lemma*.- 2.6 Circle maps.- 2.7 The problem of stability of the solar system*.- 2.8 KAM tori in perturbed Kerr blackhole*.- Chapter 3. The Hyperbolicity Theory.- 3.1 Arnold diffusion in Arnold’s example.- 3.2 Normally hyperbolic invariant manifolds(NHIM).- 3.3 The symplectic NHIM theorem.- 3.4 NHIM near resonance.- 3.5 The scattering map*.- Chapter 4. The Variational Theory.- 4.1 Tonelli Lagrangian and minimal measures.- 4.2 (Co)homology aspects of the variational principle.- 4.3 The Aubry set and the Mane set.- 4.4 The pendulum as a prototypical example.- 4.5 The Aubry sets and the α-function.- 4.6 The variational theory meets the normal form theory.- 4.7 Generic properties of minimal measures.- 4.8 The weak KAM theory.- 4.9 The hyperbolic dynamics picture of the weak KAM theory.- 4.10 Preliminary viscosity solution theory*.- 4.11 Literature review.- Chapter 5. Systems of Two Degrees of Freedom.- 5.1 Twist maps I: the structure theory.- 5.2Twist maps II: advanced topics.- 5.3 Zero energy level I: flat of α-function.- 5.4 Zero energy level II: destroy Mane sets.- 5.5. Intermediate energy levels: hyperbolic periodic orbits.- 5.6 High and low energy levels: uniform normal hyperbolicity.- 5.7 Lorentzian systems*.- Chapter 6. The Connecting Orbit Theory.- 6.1 Cohomological equivalence.- 6.2 Local connecting orbits.- 6.3 Global connecting orbits.- Part II Arnold Diffusion.- Chapter 7. Arnold Diffusion Emerging from Chaos: an Overview.- 7.1 Statements of the main theorems and their implications.- 7.2 Outline of the proofs.- 7.3 Outreaches*.- Chapter 8. Arnold Diffusion in a priori Unstable Systems.- 8.1 The big gap problem and the globalization problem.- 8.2 The globalization theory.- Chapter 9. Arnold Diffusion in Systems of Three Degrees of Freedom.- 9.1 Frequency path and three regimes.- 9.2 Normal forms, normal hyperbolicity and strong double resonances.- 9.3 Dynamics around strong double resonances.- 9.4 Proof of the main theorem in the case of n = 3.- 9.5 NHICs get arbitrarily close to the strong double resonances.- 9.6 Literature review.- Chapter 10. Arnold Diffusion in Systems of n(> 3) Degrees of Freedom.- 10.1 Superstructure of the proof.- 10.2 The first approximation of frequency segment.- 10.3 Double resonances.- 10.4 The frequency refinement.- 10.5 Triple resonances.- 10.6 Crossing triple resonances.- 10.7 Construction of the global frequency path.- 10.8 Crossing complete resonances.- A Elliptic and Hyperelliptic Curves, Abel-Jacobi Theorem*.- B Preliminary Ergodic Theory.- C Tonelli’s Theorem.- D Transversality and Genericity.- References.