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基本説明
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.
Full Description
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.
Contents
Some aspects of finite-dimensional Hamiltonian dynamics.- Four lectures on the N-body problem.- Averaging method and adiabatic invariants.- Transformation theory of Hamiltonian PDE and the problem of water waves.- Three theorems on perturbed KdV.- Groups and topology in the Euler hydrodynamics and KdV.- Infinite dimensional dynamical systems and the Navier-Stokes equation.- Hamiltonian systems and optimal control.- KAM theory with applications to Hamiltonian partial differential equations.- Four lectures on KAM for the non-linear Schrödinger equation.- A Birkhoff normal form theorem for some semilinear PDEs.- Normal form of holomorphic dynamical systems.- Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation.- Variational methods for the problem of Arnold diffusion.- The calculus of variations and the forced pendulum.- Variational methods for Hamiltonian PDEs.- Spectral gaps of potentials in weighted Sobolev spaces.- On the well-posedness of the periodic KdV equation in high regularity classes.
