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.