Symplectic Invariants and Hamiltonian Dynamics

Symplectic Invariants and Hamiltonian Dynamics
Author :
Publisher : Springer Science & Business Media
Total Pages : 353
Release :
ISBN-10 : 9783034801041
ISBN-13 : 3034801041
Rating : 4/5 (41 Downloads)

Synopsis Symplectic Invariants and Hamiltonian Dynamics by : Helmut Hofer

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

Symplectic Invariants and Hamiltonian Dynamics
Author :
Publisher : Birkhäuser
Total Pages : 356
Release :
ISBN-10 : 9783034885409
ISBN-13 : 3034885407
Rating : 4/5 (09 Downloads)

Synopsis Symplectic Invariants and Hamiltonian Dynamics by : Helmut Hofer

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.

Hamiltonian Dynamics

Hamiltonian Dynamics
Author :
Publisher : World Scientific
Total Pages : 457
Release :
ISBN-10 : 9789814496735
ISBN-13 : 9814496731
Rating : 4/5 (35 Downloads)

Synopsis Hamiltonian Dynamics by : Gaetano Vilasi

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.

Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces

Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces
Author :
Publisher : Springer Science & Business Media
Total Pages : 158
Release :
ISBN-10 : 9781461202691
ISBN-13 : 1461202698
Rating : 4/5 (91 Downloads)

Synopsis Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces by : Victor Guillemin

The action of a compact Lie group, G, on a compact sympletic manifold gives rise to some remarkable combinatorial invariants. The simplest and most interesting of these is the moment polytopes, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope. This book is addressed to researchers and can be used as a semester text.

The Breadth of Symplectic and Poisson Geometry

The Breadth of Symplectic and Poisson Geometry
Author :
Publisher : Springer Science & Business Media
Total Pages : 666
Release :
ISBN-10 : 9780817644192
ISBN-13 : 0817644199
Rating : 4/5 (92 Downloads)

Synopsis The Breadth of Symplectic and Poisson Geometry by : Jerrold E. Marsden

* The invited papers in this volume are written in honor of Alan Weinstein, one of the world’s foremost geometers * Contributions cover a broad range of topics in symplectic and differential geometry, Lie theory, mechanics, and related fields * Intended for graduate students and working mathematicians, this text is a distillation of prominent research and an indication of future trends in geometry, mechanics, and mathematical physics

Lectures on Symplectic Geometry

Lectures on Symplectic Geometry
Author :
Publisher : Springer
Total Pages : 240
Release :
ISBN-10 : 9783540453307
ISBN-13 : 354045330X
Rating : 4/5 (07 Downloads)

Synopsis Lectures on Symplectic Geometry by : Ana Cannas da Silva

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Function Theory on Symplectic Manifolds

Function Theory on Symplectic Manifolds
Author :
Publisher : American Mathematical Soc.
Total Pages : 282
Release :
ISBN-10 : 9781470416935
ISBN-13 : 147041693X
Rating : 4/5 (35 Downloads)

Synopsis Function Theory on Symplectic Manifolds by : Leonid Polterovich

This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.

Symplectic Geometry and Topology

Symplectic Geometry and Topology
Author :
Publisher : American Mathematical Soc.
Total Pages : 452
Release :
ISBN-10 : 0821886894
ISBN-13 : 9780821886892
Rating : 4/5 (94 Downloads)

Synopsis Symplectic Geometry and Topology by : Yakov Eliashberg

Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introductionto Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristicsand Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Symplectic Topology and Measure Preserving Dynamical Systems

Symplectic Topology and Measure Preserving Dynamical Systems
Author :
Publisher : American Mathematical Soc.
Total Pages : 192
Release :
ISBN-10 : 9780821848920
ISBN-13 : 0821848925
Rating : 4/5 (20 Downloads)

Synopsis Symplectic Topology and Measure Preserving Dynamical Systems by : Albert Fathi

The papers in this volume were presented at the AMS-IMS-SIAM Joint Summer Research Conference on Symplectic Topology and Measure Preserving Dynamical Systems held in Snowbird, Utah in July 2007. The aim of the conference was to bring together specialists of symplectic topology and of measure preserving dynamics to try to connect these two subjects. One of the motivating conjectures at the interface of these two fields is the question of whether the group of area preserving homeomorphisms of the 2-disc is or is not simple. For diffeomorphisms it was known that the kernel of the Calabi invariant is a normal proper subgroup, so the group of area preserving diffeomorphisms is not simple. Most articles are related to understanding these and related questions in the framework of modern symplectic topology.

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory
Author :
Publisher : American Mathematical Soc.
Total Pages : 282
Release :
ISBN-10 : 9781470436254
ISBN-13 : 1470436256
Rating : 4/5 (54 Downloads)

Synopsis Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory by : Kenji Fukaya

In this paper the authors first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating bulk deformations, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher in a slightly less general context. Then the authors explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasi-morphisms and new Lagrangian intersection results on toric and non-toric manifolds. The most novel part of this paper is its use of open-closed Gromov-Witten-Floer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasi-morphism to the Lagrangian Floer theory (with bulk deformation). The authors use this open-closed Gromov-Witten-Floer theory to produce new examples. Using the calculation of Lagrangian Floer cohomology with bulk, they produce examples of compact symplectic manifolds which admits uncountably many independent quasi-morphisms . They also obtain a new intersection result for the Lagrangian submanifold in .