Symplectic Invariants and Halmiltonian DynamicsThe 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. |
Contents
I | 1 |
III | 6 |
IV | 9 |
V | 18 |
VI | 23 |
VII | 31 |
VIII | 35 |
IX | 42 |
XXV | 161 |
XXVI | 165 |
XXVII | 173 |
XXVIII | 182 |
XXIX | 193 |
XXX | 194 |
XXXI | 202 |
XXXII | 217 |
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Common terms and phrases
1-periodic Alexander-Spanier cohomology Arnold conjecture Assume assumption Axiom boundary bounded sets bundle capacity function Chapter closed characteristic co(U cohomology compact symplectic manifold conclude Consequently constant contact type convex coordinates critical points defined definition denote diffeomorphism differential equations estimate exists finite fixed points Floer homology flow Fredholm function H geodesic Geometry gradient equation Hamiltonian function Hamiltonian system Hamiltonian vector field hence Hofer holomorphic homeomorphism homotopy hypersurface Lemma loop Math metric monotonicity Moreover Morse nondegenerate open neighborhood open set periodic orbits periodic solutions proof of Theorem Proposition prove R²n Recall regular energy surface Riemannian Riemannian metric Rx S¹ satisfying sequence smooth function subset symplectic diffeomorphism symplectic embedding symplectic invariants symplectic manifold symplectic map symplectic structure theory topology torus V₁ vector field XH VH(x ди მა