| Description |
1 online resource (xi, 115 pages) |
| Series |
Mathematical notes ; 50 |
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Mathematical notes (Princeton University Press) ; 50.
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| Contents |
880-01 3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians4 Action-Minimizing Curves for Tonelli Lagrangians; 4.1 Global Action-Minimizing Curves: Aubry and Mañé Sets; 4.2 Some Topological and Symplectic Properties of the Aubry and Mañé Sets; 4.3 An Example: The Simple Pendulum (Part II); 4.4 Mather's Approach: Peierls' Barrier; 5 The Hamilton-Jacobi Equation and Weak KAM Theory; 5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi's Weak KAM theory; 5.2 Regularity of Critical Subsolutions; 5.3 Non-Wandering Points of the Mañé Set; Appendices |
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880-01/(S Cover; Copyright; Title; Contents; Preface; 1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds; 1.1 Lagrangian Point of View; 1.2 Hamiltonian Point of View; 2 From KAM Theory to Aubry-Mather Theory; 2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori; 3 Action-Minimizing Invariant Measures for Tonelli Lagrangians; 3.1 Action-Minimizing Measures and Mather Sets; 3.2 Mather Measures and Rotation Vectors; 3.3 Mather's α- and β-Functions ; 3.4 The Symplectic Invariance of Mather Sets; 3.5 An Example: The Simple Pendulum (Part I) |
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A On the Existence of Invariant Lagrangian GraphsA. 1 Symplectic Geometry of the Phase Space; A.2 Existence and Nonexistence of Invariant Lagrangian Graphs; B Schwartzman Asymptotic Cycle and Dynamics; B.1 Schwartzman Asymptotic Cycle; B.2 Dynamical Properties; Bibliography; Index |
| 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 |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Hamiltonian systems.
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Hamilton-Jacobi equations.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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MATHEMATICS -- General.
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Hamilton-Jacobi equations
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Hamiltonian systems
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| Form |
Electronic book
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| ISBN |
9781400866618 |
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1400866618 |
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