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Symplectic Invariants and Hamiltonian Dynamics

2011-03-31 Mathematics
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Release : 2011-03-31
Genre : Mathematics
Kind : eBook
Book Rating : 041/5 ( reviews)

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Book Synopsis Symplectic Invariants and Hamiltonian Dynamics by : Helmut Hofer

Download or read book Symplectic Invariants and Hamiltonian Dynamics written by Helmut Hofer. This book was released on 2011-03-31. Available in PDF, EPUB and Kindle. Book excerpt: The discoveries of the last decades have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.

Symplectic Invariants and Hamiltonian Dynamics

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 407/5 ( reviews)

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Book Synopsis Symplectic Invariants and Hamiltonian Dynamics by : Helmut Hofer

Download or read book Symplectic Invariants and Hamiltonian Dynamics written by Helmut Hofer. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: Analysis of an old variational principal in classical mechanics has established global periodic phenomena in Hamiltonian systems. One of the links is a class of sympletic invariants, called sympletic capacities, and these invariants are the main theme of this book. Topics covered include basic sympletic geometry, sympletic capacities and rigidity, sympletic fixed point theory, and a survey on Floer homology and sympletic homology.

Lectures on Dynamical Systems

2010 Dynamics
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Release : 2010
Genre : Dynamics
Kind : eBook
Book Rating : 814/5 ( reviews)

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Book Synopsis Lectures on Dynamical Systems by : Eduard Zehnder

Download or read book Lectures on Dynamical Systems written by Eduard Zehnder. This book was released on 2010. Available in PDF, EPUB and Kindle. Book excerpt: This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at ETH Zurich. The first part centers around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale's theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum. The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.

Hamiltonian Dynamics

2001 Mathematics
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Release : 2001
Genre : Mathematics
Kind : eBook
Book Rating : 311/5 ( reviews)

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Book Synopsis Hamiltonian Dynamics by : Gaetano Vilasi

Download or read book Hamiltonian Dynamics written by Gaetano Vilasi. This book was released on 2001. Available in PDF, EPUB and Kindle. Book excerpt: This is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems. As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity. As a monograph, the book deals with the advanced research topic of completely integrable dynamics, with both finitely and infinitely many degrees of freedom, including geometrical structures of solitonic wave equations. Contents: Analytical Mechanics: The Lagrangian Coordinates; Hamiltonian Systems; Transformation Theory; The Integration Methods; Basic Ideas of Differential Geometry: Manifolds and Tangent Spaces; Differential Forms; Integration Theory; Lie Groups and Lie Algebras; Geometry and Physics: Symplectic Manifolds and Hamiltonian Systems; The Orbits Method; Classical Electrodynamics; Integrable Field Theories: KdV Equation; General Structures; Meaning and Existence of Recursion Operators; Miscellanea; Integrability of Fermionic Dynamics. Readership: Physicists and mathematicians.

Symplectic Topology and Floer Homology: Volume 2, Floer Homology and its Applications

2015-08-27 Mathematics
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Release : 2015-08-27
Genre : Mathematics
Kind : eBook
Book Rating : 390/5 ( reviews)

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Book Synopsis Symplectic Topology and Floer Homology: Volume 2, Floer Homology and its Applications by : Yong-Geun Oh

Download or read book Symplectic Topology and Floer Homology: Volume 2, Floer Homology and its Applications written by Yong-Geun Oh. This book was released on 2015-08-27. Available in PDF, EPUB and Kindle. Book excerpt: Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floer theory as a whole. Volume 2 provides a comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer theory, including many examples of their applications to various problems in symplectic topology. The first volume covered the basic materials of Hamiltonian dynamics and symplectic geometry and the analytic foundations of Gromov's pseudoholomorphic curve theory. Symplectic Topology and Floer Homology is a comprehensive resource suitable for experts and newcomers alike.

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