Sorrentino Action-minimizing Methods in Hamiltonian Dynamics

An Introduction to Aubry-Mather Theory
Pilot project, eBook available to selected US libraries only
ISBN: 978-1-4008-6661-8
Verlag: De Gruyter
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 50, 128 Seiten

Reihe: Mathematical Notes

No detailed description available for "Action-minimizing Methods in Hamiltonian Dynamics".

Autoren/Hrsg.

Sorrentino, Alfonso

Inhaltsverzeichnis

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

Über Autor(innen)

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.