Symplectic Invariants and Hamiltonian Dynamics (Modern Birkhäuser Classics) by Helmut Hofer (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I would guess this is still the best place to learn the Hofer-Zehnder capacity and Hofer’s metric. The proof of the Arnold conjecture for the torus and gentle approach to Floer/symplectic homology are beautifully done as well. A clear, rigorous introduction to Hamiltonian dynamics from the perspective of symplectic topology, suitable for beginning graduate/advanced undergraduate students.Unfortunately, are no exercises. Also, I think it’s usefulness as a reference is limited (I would not change this, because this is part of why it’s such a good text, but I think it’s worth mentioning).

⭐This monograph gives a good overview of a very important subject in both mathematics and physics. The phase space formulation of classical mechanics via the Hamiltonian formalism is represented throughout this book in the language of symplectic manifolds. The geometry of symplectic manifolds has many surprises considering the very simple definition of a symplectic form. One would not have suspected that they would imply the Gromov squeezing theorem and other interesting results. The authors do an excellent job of explaining this theorem and other intricacies of symplectic geometry, in particular the idea of symplectic capacities, Floer-Morse theory, and the Arnold conjecture. The Gromov squeezing theorem has an analogue in quantum physics. Jokingly called “quantum claustrophobia” by physicists, this represents the idea that particles in the quantum realm tend to avoid small regions of phase space. The book should be of great assistance to the mathematically-inclined physicist interested in the properties of nonlinear dynamical systems in phase space. Even though the KAM theory is only briefly discussed in the book, the presentation should prepare the physicist reader for additional reading on the subject. The most interesting and important part of the book is the discussion on how symplectic capacities are related to volumes and Lebesgue measures in ordinary Euclidean space. The capacity and volume agree as invariants only for two-dimensional symplectic manifolds. In addition, in four dimensions and higher, open sets can be very different in terms of shape, size, measure, and topology, and still have the same capacity. The author’s presentation of these facts is very well-written and thought provoking. Probably the most difficult part of the book is the justification of the squeezing theorem using variational principles. The technical details of the proof are very intricate and take lots of time to digest. Everything about the symplectic category is interesting and this book is a fine compendium of modern results in the field.

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