Preface.- Hamiltonian Dynamical Systems and Applications; W. Craig.- Some Aspects of Finite-Dimensional Hamiltonian Dynamics; D.V. Treschev.- Four Lectures on the N-body Problem; A. Chenciner.- Average Method and Adiabatic Invariants; A. Neishtadt.- Transformation Theory of Hamiltonian PDE and the Problem of Water Waves; W. Craig.- Three Theorems on Perturbed KdV; S.B. Kuksin.- Groups and Topology in the Euler Hydrodynamics and KdV; B. Khesin.- Infinite Dimensional Dynamical Systems and the Navier-Stokes Equation; C.E. Wayne.- Hamiltonian Systems and Optimal Control; A. Agrachev.- KAM Theory with Applications to Hamiltonian Partial Differential Equations; X. Yuan.- Four Lectures on KAM for the Non-Linear Schroedinger Equation; L.H. Eliasson, S.B. Kuksin.- A Birkhoff Normal Form Theorem for some Semilinear PDEs; D. Bambusi.- Normal Forms for Holomorphic Dynamical Systems; L. Stolovitch.- Geometric Approaches to the Problem of Instability in Hamiltonian Systems. An Informal Presentation; A. Delshams et al.- Variational Methods for the Problem of Arnold Diffusion; C.-Q. Cheng.- The Calculus of Variations and the Forced Pendulum; P.H. Rabinowitz.- Variational Methods for Hamiltonian PDEs; M. Berti.- Spectral Gaps of Potentials in Weighted Sobolev Spaces; J. Poeschel.- On the Well-Posedness of the Periodic KdV Equation in High Regularity Classes; T. Kappeler, J. Poeschel.
This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.
Lecture notes on current state-of-the-art by the researchers who have developed the theory
Introductions of the technically deep methods of Hamiltonian mechanics to partial differential equations
Contains lectures on the most recent advances in KAM theory for partial differential equations
Contains a discursive but complete description of the two main programs of study of Arnold diffusion, the first such in the literature
Gives an introduction to the theory of Hamiltonian PDE, arguably one of the most dynamic fields of PDE today, that is accessible to graduate students and mathematical researchers outside of the discipline