Hamiltonian dynamical systems and applications: [proceedings of the NATO Advanced Study Institute on Hamiltonian Dynamical Systems and Applications, Montreal, Canada, 18-29 June 2007]

Physical laws are for the most part expressed in terms of differential equations, and natural classes of these are in the form of conservation laws or of problems of the calculus of variations for an action functional. These problems can generally be posed as Hamiltonian systems, whether dynamical systems on finite dimensional phase space as in classical mechanics, or partial differential equations (PDE) which are naturally of infinitely many degrees of freedom. 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 as well as the theory of Hamiltonian systems in infinite dimensional phase space; these are described in depth in this volume. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations. These lecture notes cover many areas of recent mathematical progress in this field, including the new choreographies of many body orbits, the development of rigorous averaging methods which give hope for realistic long time stability results, the development of KAM theory for partial differential equations in one and in higher dimensions, and the new developments in the long outstanding problem of Arnold diffusion. It also includes other contributions to celestial mechanics, to control theory, to partial differential equations of fluid dynamics, and to the theory of adiabatic invariants. In particular the last several years has seen major progress on the problems of KAM theory and Arnold diffusion; accordingly, this volume includes lectures on recent developments of KAM theory in infinite dimensional phase space, and descriptions of Arnold diffusion using variational methods as well as geometrical approaches to the gap problem. The subjects in question involve by necessity some of the most technical aspects of analysis coming from a number of diverse fields. Before the present volume, there has not been one text nor one course of study in which advanced students or experienced researchers from other areas can obtain an overview and background to enter this research area. This volume offers this, in an unparalleled series of extended lectures encompassing this wide spectrum of topics in PDE and dynamical systems. Some aspects of finite-dimensional Hamiltonian dynamics / D.V. Treschev Four lectures on the N-body problem / Alain Chenciner Averaging method and adiabatic invariants / Anatoly Neishtadt Transformation theory of Hamiltonian PDE and the problem of water waves / Walter Craig Three theorems on perturbed KdV / Sergei B. Kuksin Groups and topology in the Euler hydrodynamics and KdV / Boris Khesin Infinite dimensional dynamical systems and the Navier-Stokes equation / C. Eugene Wayne Hamiltonian systems and optimal control / Andrei Agrachev KAM theory with applications to Hamiltonian partial differential equations / Xiaoping Yuan Four lectures on KAM for the non-linear Schrödinger equation / L.H. Eliasson and Sergei B. Kuksin A Birkhoff normal form theorem for some semilinear PDEs / D. Bambusi Normal form of holomorphic dynamical systems / Laurent Stolovitch Geometric approaches to the problem of instability in Hamiltonian systems : an informal presentation / Amadeu Delshams, Marian Gidea, Rafael de la Llave and Tere M. Seara Variational methods for the problem of Arnold diffusion / Chong-Qing Cheng The calculus of variations and the forced pendulum / Paul H. Rabinowitz Variational methods for Hamiltonian PDEs / Massimiliano Berti Spectral gaps of potentials in weighted Sobolev spaces / Jürgen Pöschel On the well-posedness of the periodic KdV equation in high regularity classes / Thomas Kappeler and Jürgen Pöschel.