Symplectic Invariants and Hamiltonian Dynamics

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.

• Symplectic topology
• action principle
• capacities
• symplectic invariants
• symplectic manifolds

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)

• , School of Mathematics, Institute for Advanced Study (IAS), Princeton, USA

  Helmut Hofer

• , Departement Mathematik, ETH Zürich, Zürich, Switzerland

  Eduard Zehnder

Helmut Hofer is Professor at the Institute for Advanced Study in Princeton, New Jersey, USA.

Eduard Zehnder is emeritus Professor of ETH Zürich, Switzerland.

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