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Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory

Alfonso Sorrentino
Publisher: 
Princeton University Press
Publication Date: 
2015
Number of Pages: 
115
Format: 
Paperback
Series: 
Mathematical Notes
Price: 
45.00
ISBN: 
9780691164502
Category: 
Monograph
We do not plan to review this book.

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