Overview

The discoveries of the last decade have shed new light on the old field of Hamiltonian systems and have 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. 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 related. One of the links is a class of symplectic invariants, called symplectic capacitites. These invariants are the main topic of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the least action principle, a bi-variant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed-point theory and first-order elliptic systems, the Arnold conjectures and a survey on Floer - and symplectic homology. The exposition is self-contained and addressed to researchers and students from the graduate level onwards.

Full Product Details

Author:   Helmut Hofer ,  Eduard Zehnder
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Dimensions:   Width: 16.50cm , Height: 2.30cm , Length: 24.00cm
Weight:   0.820kg
ISBN:  

9783764350666

ISBN 10:   3764350660
Pages:   356
Publication Date:   01 August 1994
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Out of Print
Availability:   Out of stock  

Table of Contents

Part 1 Introduction: symplectic vectro spaces; symplectic diffeomorphisms and Hamiltonian vector fields in (R2n, omega-0); Hamiltonian vector fields and symplectic manifolds; periodic orbits on energy surfaces; existence of a periodic orbit on a convex energy surface; the problem of symplectic embeddings; symplectic classification of positive definite quadratic forms; the orbit structure near an equilibrium, Birkhoff normal form. Part 2 Symplectic capacities: definition and application to embeddings; rigidity of symplectic diffeomorphisms. Part 3 Existence of a capacity: definition of the capacity c-0; the minimax idea; the anlytical setting; the existence of a critical point; examples and illustrations. Part 4 Existence of closed characteristics: periodic solutions in energy surfaces; the characterisrtic line bundle of a hypersurface; hypersurfaces of contact type, the Weinstein conjecture; classical Hamiltonian systems; the torus and Herman's non-closing Lemma. Part 5 Compactly supported symplectic mappings: a special metric d for a group D ; the action spectrum of a Hamiltonian map; a universal variational principle; a continuous section of the action spectrum bundle; an inequality between the displacement energy; comparison of the metric d on D with the C0-metric ; fixed points and geodesics on D . Part 6 The Arnold conjecture, Floer homology: the Arnold conjecture on symplectic fixed points; the model case of the torus; gradient-like flows on compact spaces; elliptic methods and symplectic fixed points; Floer's approach to Morse theory for the action functional; symplectic homology; generating functions of symplectic mappings in R2n; action-angle coordinates, the theorem of Arnold and Jost; embeddings of H1/2(S1) and smoothness of the action; the Cauchy-Riemann operator on the sphere; ellpitic estimates near the boundary and an application; the generalized Carleman similarity principle; the Brouwer degree; continuity property of the Alexander-Spanier cohomology.

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