Action-minimizing methods in Hamiltonian dynamics : an introduction to Aubry-Mather theory / Alfonso Sorrentino.

Format:

Book

Author/Creator:

Sorrentino, Alfonso, author.

Series:

Mathematical notes (Princeton University Press) ; 50.

Mathematical Notes ; 50

Language:

English

Subjects (All):

Hamiltonian systems.

Hamilton-Jacobi equations.

Physical Description:

1 online resource (129 p.)

Place of Publication:

Princeton, [New Jersey] ; Oxford, [England] : Princeton University Press, 2015.

Language Note:

English

Summary:

John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach-known as Aubry-Mather theory-singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic.Starting with the mathematical background from which Mather's theory was born, Alfonso Sorrentino first focuses on the core questions the theory aims to answer-notably the destiny of broken invariant KAM tori and the onset of chaos-and describes how it can be viewed as a natural counterpart of KAM theory. He achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. Sorrentino then describes the whole theory and its subsequent developments and applications in their full generality.Shedding new light on John Mather's revolutionary ideas, this book is certain to become a foundational text in the modern study of Hamiltonian systems.

Contents:

Front matter

Contents

Preface

Chapter One. Tonelli Lagrangians and Hamiltonians on Compact Manifolds

Chapter Two. From KAM Theory to Aubry-Mather Theory

Chapter Three. Action-Minimizing Invariant Measures for Tonelli Lagrangians

Chapter Four. Action-Minimizing Curves for Tonelli Lagrangians

Chapter Five. The Hamilton-Jacobi Equation and Weak KAM Theory

Appendices

Appendix A. On the Existence of Invariant Lagrangian Graphs

Appendix B. Schwartzman Asymptotic Cycle and Dynamics

Bibliography

Index

Notes:

Description based upon print version of record.

Pilot project,eBook available to selected US libraries only

Includes bibliographical references and index.

Description based on print version record.

OCLC:

908080604