Action-minimizing Methods in Hamiltonian Dynamics (eBook)

An Introduction to Aubry-Mather Theory

Alfonso Sorrentino (Autor)

eBook Download: PDF | EPUB

2015
128 Seiten
Princeton University Press (Verlag)
978-1-4008-6661-8 (ISBN)

Infos

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.

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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.

Produkt-Details

• Erscheint lt. Verlag: 9.6.2015
• Reihe/Serie: Mathematical Notes
• Reihe/Serie: Mathematical Notes
• Zusatzinfo: 4 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik
• Schlagworte: absolute continuity • action-minimizing measure • action-minimizing orbits • Addition • Affine space • Albert Fathi • Asymptote • Aubry set • AubryЍather theory • Betti number • Chaos • Characterization (mathematics) • classical mechanics • cohomology • Commutative diagram • compact manifold • Compact space • Conserved quantity • continuous function • convex combination • convex function • convex set • cotangent bundle • counterexample • Covering space • Derivative • Diagram (category theory) • Differentiability • Differentiable function • Dimension • Dirac measure • Division by zero • Duality (optimization) • dual space • Dynamical system • energy level • Equation • equicontinuity • Ergodicity • ergodic theory • Euler–Lagrange equation • Exact differential • existential quantification • Family of sets • fiber bundle • Floer homology • foliation • Generic property • Hamiltonian Dynamics • hamiltonian mechanics • hamiltonians • Hamiltonian System • Hamiltonian vector field • Hamilton–Jacobi equation • HamiltonЊacobi equation • Homology (mathematics) • Homomorphism • Hyperbolic 3-manifold • hypersurface • incompressible surface • Infimum and supremum • Injective function • integrable system • Integral curve • invariant Lagrangian graphs • invariant measure • invariant probability measures • invariant sets • Inverse function • John Mather • KAM Theory • KAM tori • Klein Bottle • Lagrange multiplier • Lagrangian • Lagrangian dynamics • Lagrangian (field theory) • Lagrangian system • Lebesgue measure • Legendre transformation • Limit point • Limit superior and limit inferior • Linear Function • Lipschitz continuity • Maher sets • MAK tori • Ma's critical value • Ma set • Ma's potential • Maxima and minima • Metric Space • Monotonic Function • Orbits • Orientability • Pairing • parametrization • Peierls' barrier • pendulum • periodic point • perturbation theory • Phase space • Pointwise • Poisson Bracket • probability measure • quadratic form • Quantity • Semi-continuity • Separatrix (mathematics) • Smoothness • Special case • stable motion • strict convexity • submanifold • subsequence • Subset • symmetrization • Symplectic Geometry • Symplectic Manifold • Symplectomorphism • tangent bundle • Theorem • Tonelli Lagrangians • Tonelli's theorem (functional analysis) • topological group • Topology • unstable motion • Variational method (quantum mechanics) • Variational Principle • Vector field • Vector Space • weak solution
• ISBN-10: 1-4008-6661-8 / 1400866618
• ISBN-13: 978-1-4008-6661-8 / 9781400866618