The discoveries of the past decade 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 map pings is very different from that of volume preserving mappings which raised new questions, many of them still unanswered. On the other hand, due to the analysis of an old variational principle in classical mechanics, global periodic phenomena in Hamiltonian systems have been established. As it turns out, these seemingly differ ent 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 grew out of lectures given by the authors at Rutgers University, the RUB Bochum and at the ETH Zurich (1991) and also at the Borel Seminar in Bern 1992. Since the lectures did not require any previous knowledge, only a few and rather elementary topics were selected and proved in detail. Moreover, our se lection has been prompted by a single principle: the action principle of mechanics. The action functional for loops in the phase space, given by 1 Fh) = J pdq -J H(t, 'Y(t)) dt , 'Y 0 differs from the old Hamiltonian principle in the configuration space defined by a Lagrangian. The critical points of F are those loops 'Y which solve the Hamiltonian equations associated with the Hamiltonian H and hence are the periodic orbits.

Autorentext

Helmut Hofer has contributed to nonlinear analysis, the theory of dynamical systems and symplectic geometry and topology. He is one of the founders of symplectic topology and is known for Hofer Geometry, his work on the Arnold conjectures and Weinstein conjecture, and, with various collaborators and co-authors, symplectic capacity theory, symplectic homology, symplectic field theory, finite energy foliations and their applications to dynamical systems, polyfold theory and feral curve theory. He currently holds the Hermann Weyl Professorship at the Institute for Advanced Study in Princeton. Kris Wysocki has contributed to nonlinear analysis, the theory of dynamical systems and symplectic geometry and topology. He is known as one of the originators of the theory of finite energy foliations and its applications to Hamiltonian dynamics, the compactness result of symplectic field theory, applications of symplectic homology, and polyfold theory. At the time of his passinghe was Professor at Pennsylvania State University. Eduard Zehnder is one of the founders of the field of symplectic topology. Well known are his contributions to Hamiltonian systems close to integrable ones. Jointly with C. Conley, he proved the Arnold Conjecture for symplectic fixed points on tori. This meanwhile classical result, referred to as the Conley-Zehnder Theorem, together with Gromov's pseudoholomorphic curve theory led Zehnder's student Andreas Floer to introduce the seminal concept of Floer Homology. With H. Hofer and K. Wysocki, he worked on global periodic phenomena in Hamiltonian and Reeb dynamics, compactness problems in symplectic field theory and on the theory and applications of polyfolds. He is currently Professor Emeritus at ETH Zurich.

Inhalt
1 Introduction.- 1.1 Symplectic vector spaces.- 1.2 Symplectic diffeomorphisms and Hamiltonian vector fields.- 1.3 Hamiltonian vector fields and symplectic manifolds.- 1.4 Periodic orbits on energy surfaces.- 1.5 Existence of a periodic orbit on a convex energy surface.- 1.6 The problem of symplectic embeddings.- 1.7 Symplectic classification of positive definite quadratic forms.- 1.8 The orbit structure near an equilibrium, Birkhoff normal form.- 2 Symplectic capacities.- 2.1 Definition and application to embeddings.- 2.2 Rigidity of symplectic diffeomorphisms.- 3 Existence of a capacity.- 3.1 Definition of the capacity c0.- 3.2 The minimax idea.- 3.3 The analytical setting.- 3.4 The existence of a critical point.- 3.5 Examples and illustrations.- 4 Existence of closed characteristics.- 4.1 Periodic solutions on energy surfaces.- 4.2 The characteristic line bundle of a hypersurface.- 4.3 Hypersurfaces of contact type, the Weinstein conjecture.- 4.4 Classical Hamiltonian systems.- 4.5 The torus and Herman's Non-Closing Lemma.- 5 Compactly supported symplectic mappings in ?2n.- 5.1 A special metric d for a group D of Hamiltonian diffeomorphisms.- 5.2 The action spectrum of a Hamiltonian map.- 5.3 A universal variational principle.- 5.4 A continuous section of the action spectrum bundle.- 5.5 An inequality between the displacement energy and the capacity.- 5.6 Comparison of the metric d on D with the C0-metric.- 5.7 Fixed points and geodesics on D.- 6 The Arnold conjecture, Floer homology and symplectic homology.- 6.1 The Arnold conjecture on symplectic fixed points.- 6.2 The model case of the torus.- 6.3 Gradient-like flows on compact spaces.- 6.4 Elliptic methods and symplectic fixed points.- 6.5 Floer's appraoch to Morse theory for the actionfunctional.- 6.6 Symplectic homology.- A.2 Action-angle coordinates, the Theorem of Arnold and Jost.- A.4 The Cauchy-Riemann operator on the sphere.- A.5 Elliptic estimates near the boundary and an application.- A.6 The generalized similarity principle.- A.7 The Brouwer degree.- A.8 Continuity property of the Alexander-Spanier cohomology.