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.
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 Poincares 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 Arnolds 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 Tonellis Theorem.- D Transversality and Genericity.- References.
Chongqing Cheng is a professor at Nanjing University. He has been engaged in the research of dynamical systems for many years. His main research interest is Hamiltonian dynamical systems, including dynamical instability, variational construction of connecting orbits, Arnold diffusion, KAM theory and weak KAM theory. He was an invited speaker at ICM 2010 Hyderabad.
Jinxin Xue is a professor at Tsinghua University. His field of research is dynamical systems and his primary interests are to find various special orbits in dynamical systems. Prof. Xue solved the Painleve conjecture and the Arnold diffusion conjecture (jointly with C-Q. Cheng) in the smooth category for convex systems. Besides Hamiltonian dynamics, he also works on fields including hyperbolic dynamics, group actions, symplectic topology, mean curvature flow, etc.
