Description
Book SynopsisJohn 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 posse
Table of ContentsPreface 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 Mane Sets 48 4.2 Some Topological and Symplectic Properties of the Aubry and Mane 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 Mane 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
