Synopses & Reviews
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.
Review
The Aubry-Mather approach to the study of the dynamics of twistdiffeomorphisms of the annulus--corresponding to Poincar� maps of one-dimensional Hamiltonian systems--reveals the existence of manyaction-minimizing sets, says Sorrentino, which in some sense generalize invariant rotational curves and which always exist, evenafter rotational curves are destroyed. He covers Tonelli Lagrangians and Hamiltonians on compact manifolds, from KAM theory toAubry-Mather theory, action-minimizing invariant measures for Tonelli Lagrangians, action-minimizing curves for Tonelli Lagrangians, and the Hamtonian-Jacobi equation and weak KAM theory.Annotation �2015 Ringgold, Inc., Portland, OR (protoview.com)
Review
The Aubry-Mather approach to the study of the dynamics of twistdiffeomorphisms of the annulus--corresponding to Poincaré maps of one-dimensional Hamiltonian systems--reveals the existence of manyaction-minimizing sets, says Sorrentino, which in some sense generalize invariant rotational curves and which always exist, evenafter rotational curves are destroyed. He covers Tonelli Lagrangians and Hamiltonians on compact manifolds, from KAM theory toAubry-Mather theory, action-minimizing invariant measures for Tonelli Lagrangians, action-minimizing curves for Tonelli Lagrangians, and the Hamtonian-Jacobi equation and weak KAM theory.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
Review
The Aubry-Mather approach to the study of the dynamics of twistdiffeomorphisms of the annulus--corresponding to Poincaré maps of one-dimensional Hamiltonian systems--reveals the existence of manyaction-minimizing sets, says Sorrentino, which in some sense generalize invariant rotational curves and which always exist, evenafter rotational curves are destroyed. He covers Tonelli Lagrangians and Hamiltonians on compact manifolds, from KAM theory toAubry-Mather theory, action-minimizing invariant measures for Tonelli Lagrangians, action-minimizing curves for Tonelli Lagrangians, and the Hamtonian-Jacobi equation and weak KAM theory.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
Review
The Aubry-Mather approach to the study of the dynamics of twistdiffeomorphisms of the annulus--corresponding to Poincaré maps of one-dimensional Hamiltonian systems--reveals the existence of manyaction-minimizing sets, says Sorrentino, which in some sense generalize invariant rotational curves and which always exist, evenafter rotational curves are destroyed. He covers Tonelli Lagrangians and Hamiltonians on compact manifolds, from KAM theory toAubry-Mather theory, action-minimizing invariant measures for Tonelli Lagrangians, action-minimizing curves for Tonelli Lagrangians, and the Hamtonian-Jacobi equation and weak KAM theory.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
Review
The Aubry-Mather approach to the study of the dynamics of twistdiffeomorphisms of the annulus--corresponding to Poincaré maps of one-dimensional Hamiltonian systems--reveals the existence of manyaction-minimizing sets, says Sorrentino, which in some sense generalize invariant rotational curves and which always exist, evenafter rotational curves are destroyed. He covers Tonelli Lagrangians and Hamiltonians on compact manifolds, from KAM theory toAubry-Mather theory, action-minimizing invariant measures for Tonelli Lagrangians, action-minimizing curves for Tonelli Lagrangians, and the Hamtonian-Jacobi equation and weak KAM theory.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
Review
The Aubry-Mather approach to the study of the dynamics of twistdiffeomorphisms of the annulus--corresponding to Poincaré maps of one-dimensional Hamiltonian systems--reveals the existence of manyaction-minimizing sets, says Sorrentino, which in some sense generalize invariant rotational curves and which always exist, evenafter rotational curves are destroyed. He covers Tonelli Lagrangians and Hamiltonians on compact manifolds, from KAM theory toAubry-Mather theory, action-minimizing invariant measures for Tonelli Lagrangians, action-minimizing curves for Tonelli Lagrangians, and the Hamtonian-Jacobi equation and weak KAM theory.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
About the Author
Alfonso Sorrentino is associate professor of mathematics at the University of Rome "Tor Vergata" in Italy. He holds a PhD in mathematics from Princeton University.
Table of Contents
Preface vii
1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds 1
1.1 Lagrangian Point of View 1
1.2 Hamiltonian Point of View 4
2 From KAM Theory to Aubry-Mather Theory 8
2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori 8
3 Action-Minimizing Invariant Measures for Tonelli Lagrangians 18
3.1 Action-Minimizing Measures and Mather Sets 18
3.2 Mather Measures and Rotation Vectors 24
3.3 Mather's a-and B-Functions 28
3.4 The Symplectic Invariance of Mather Sets 35
3.5 An Example: The Simple Pendulum (Part I) 39
3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians 45
4 Action-Minimizing Curves for Tonelli Lagrangians 48
4.1 Global Action-Minimizing Curves: Aubry and Mañé Sets 48
4.2 Some Topological and Symplectic Properties of the Aubry and Mañé Sets 66
4.3 An Example: The Simple Pendulum (Part II) 68
4.4 Mather's Approach: Peierls' Barrier 71
5 The Hamilton-Jacobi Equation and Weak KAM Theory 76
5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi's Weak KAM theory 76
5.2 Regularity of Critical Subsolutions 85
5.3 Non-Wandering Points of the Mañé Set 87
Appendices A On the Existence of Invariant Lagrangian Graphs 89
A.1 Symplectic Geometry of the Phase Space 89
A.2 Existence and Nonexistence of Invariant Lagrangian Graphs 91
B Schwartzman Asymptotic Cycle and Dynamics 97
B.1 Schwartzman Asymptotic Cycle 97
B.2 Dynamical Properties 99
Bibliography 107
Index 113