Leseprobe
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Action-minimizing Methods in Hamiltonian Dynamics (MN-50)
Alfonso Sorrentino
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Cotangent bundle, Hamiltonian vector field, Lipschitz continuity, Dimension, Convex function, Subset, Ergodicity, Duality (optimization), Foliation, Perturbation theory, Lagrangian, Pairing, Quantity, Theorem, Limit point, Phase space, Dual space, Diagram (category theory), Integral curve, Smoothness, Counterexample, Tonelli's theorem (functional analysis), Symplectic geometry, Variational principle, Euler–Lagrange equation, Integrable system, Legendre transformation, Compact space, Absolute continuity, Dynamical system, Quadratic form, Homology (mathematics), Hamiltonian system, Lagrangian system, Topology, Monotonic function, Poisson bracket, Covering space, Dirac measure, Semi-continuity, Injective function, Lagrangian (field theory), Hamilton–Jacobi equation, Asymptote, Vector field, Convex combination, Symplectic manifold, Invariant measure, Separatrix (mathematics), Cohomology, Weak solution, Vector space, Subsequence, Infimum and supremum, Probability measure, Existential quantification, Addition, Energy level, Equation, Maxima and minima, Variational method (quantum mechanics), Commutative diagram, Fiber bundle, Hamiltonian mechanics, Differentiable function, Division by zero, Floer homology, Symmetrization, Topological group, Tangent bundle