Helmut Hofer, Eduard Zehnder

Symplectic Invariants and Hamiltonian Dynamics

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• Birkhäuser Advanced Texts Basler Lehrbücher
• Verlag: Birkhäuser / Birkhäuser Basel / Springer, Basel
• Artikelnr. des Verlages: 978-3-0348-9671-9
• 1994
• Seitenzahl: 364
• Erscheinungstermin: 13. Oktober 2012
• Englisch
• Abmessung: 254mm x 178mm x 20mm
• Gewicht: 685g
• ISBN-13: 9783034896719
• ISBN-10: 3034896719
• Artikelnr.: 41322104

Inhaltsangabe

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.

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1 Introduction.- 2 Symplectic capacities.- 3 Existence of a capacity.- 4 Existence of closed characteristics.- 5 Compactly supported symplectic mappings in R2n.- 6 The Arnold conjecture, Floer homology and symplectic homology.- Appendix.- Index.- Bibliography.

Rezensionen

From the reviews:

"Symplectic Invariants and Hamiltonian Dynamics is obviously a work of central importance in the field and is required reading for all would-be players in this game. Happily, it is very well written and sports a lot of very useful commentary by the authors; the sections introducing the individual chapters are particularly well done ... . It is all fine scholarship in an exciting and fertile area." (Michael Berg, The Mathematical Association of America, June, 2011)