Symplectic invariants and Hamiltonian dynamics
Author(s)
Bibliographic Information
Symplectic invariants and Hamiltonian dynamics
（Birkhäuser advanced texts : Basler Lehrbücher / edited by Herbert Amann, Hanspeter Kraft）
Birkhäuser Verlag, c1994
- : sz
- : us
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Note
Includes bibliography (p. 327-341) and index
Description and Table of Contents
Description
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: sympletic topology. Surprising rigidity phenomena demonstrate that the nature of sympletic 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. One of the links is a class of sympletic invariants, called sympletic capacities. These invariants are the main theme of this book, which includes such topics as basic sympletic geometry, sympletic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the sympletic diffeomorphism group and its geometry, sympletic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and sympletic homology.
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.
by "Nielsen BookData"
