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Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities

European Mathematical Society
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The modern study of dynamics began with difficult questions about celestial mechanics, specifically about the movement of planets and the moon. Is the solar system stable? What happens to periodic orbits when they are perturbed? How do we explain those mysterious gaps in the asteroid belt? Starting (arguably) with Poincaré, the modern study of dynamics has continued to advance and evolve with the work of Arnold, Moser, Kolmogorov, Smale and many others, including the author of this book, Eduard Zehnder.

The text itself originated with a course for third year undergraduates in mathematics and physics at ETH in Zűrich. In the U.S., this book would be most appropriate for first or second year graduate students. Prerequisites include familiarity with ordinary differential equations, the topology of metric spaces, some comfort with measure theory and the Lebesgue integral, and other incidental items like the implicit function theorem in Banach space and a bit of ergodic theory. Zehnder says that he has tried to keep the requirements minimal and that he has provided “detailed (sometimes excruciatingly detailed) proofs.” Of course, what is excruciating to the author is often a godsend to the student.

Roughly the first two-thirds of the book is devoted to topics that are now standard parts of the study of dynamical systems. The introduction emphasizes discrete dynamical systems and includes examples of minimal, transitive, structurally stable and ergodic systems. Following that, two chapters concentrate on unstable phenomena associated with a hyperbolic fixed point of a diffeomorphism; here we see homoclinic points, invariant hyperbolic sets and chaotic orbit structures. A key result is Smale’s theorem on the embedding of Bernoulli systems near homoclinic orbits, proved using the Shadowing Lemma. The following two chapters first introduce smooth flows generated by vector fields and continuous flows on metric spaces and then concentrate on Hamiltonian vector fields. Along the way we’re introduced to several important ideas: the gradient flow, Morse theory, symplectic manifolds and enough exterior calculus to develop the modern Hamiltonian formalism. Throughout there is a continuing interplay between the unstable-chaotic and the stable-regular that characterizes the study of dynamical systems and makes it both fascinating and a little crazy.

The rest of the book introduces some topics from symplectic geometry. In particular, a class of symplectic invariants called symplectic capacities provides tools for establishing results about symplectic rigidity such as the Gromov’s “non-squeezing” theorem. The final chapter uses a dynamical version of symplectic capacity due to Hofer and Zehnder to find periodic orbits of a Hamiltonian vector field on a prescribed compact energy surface. So far as I know, these results have not appeared before in a textbook at this level.

Zehnder’s book provides a thorough grounding in dynamical systems and offers a bit more than a glimpse at some applications of symplectic geometry. It would be an excellent text for a graduate course.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Date Received: 
Monday, June 28, 2010
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Eduard Zehnder
EMS Textbooks in Mathematics 11
Publication Date: 
William J. Satzer
  • Introduction
  • Invariant manifolds of hyperbolic fixed points
  • Hyperbolic sets
  • Gradientlike flows
  • Hamiltonian vector fields and symplectic diffeomorphisms
  • Questions, phenomena, results
  • Symplectic invariants
  • Applications of the capacity $c_0$ in Hamiltonian systems
  • Bibliography
  • List of symbols
  • Index
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Thursday, August 26, 2010