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: symplectic topology. Surprising rigidity phenomena
demonstrate that the nature of symplectic mappings is very
different from that of volume preserving mappings. This raises new
questions, many of them still unanswered. 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 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 includes such topics as basic symplectic
geometry, symplectic capacities and rigidity, periodic orbits for
Hamiltonian systems and the action principle, a bi-invariant metric
on the symplectic diffeomorphism group and its geometry, symplectic
fixed point theory, the Arnold conjectures and first order elliptic
systems, and finally a survey on Floer homology and symplectic
homology.
The exposition is self-contained and addressed to researchers
and students from the graduate level onwards.
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