Hamiltonian Dynamical Systems and Applications

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.

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1. Ebooks list page : 1598
2. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
3. Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples (Lecture Notes in Mathematics)
4. Construction of Mappings for Hamiltonian Systems and Their Applications
5. Dynamical Systems VII: Integrable Systems. Nonholonomic Dynamical Systems (Encyclopaedia of Mathematical Sciences)
6. Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems
7. Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems
8. Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems
9. Dynamical Systems with Applications using Mathematica
10. [share_ebook] Dynamical Systems with Applications using Mathematica
11. Random Dynamical Systems: Theory and Applications
12. Piecewise-smooth Dynamical Systems: Theory and Applications
13. Control of Nonlinear Dynamical Systems: Methods and Applications
14. Dynamical Systems VIII: Singularity Theory II. Applications
15. Polynomial and Rational Matrices: Applications in Dynamical Systems Theory
16. [share_ebook] Control of Nonlinear Dynamical Systems: Methods and Applications