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Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities
Eduard Zehnder, ETH Zurich, Switzerland
A publication of the European Mathematical Society.
cover
EMS Textbooks in Mathematics
2010; 363 pp; hardcover
Volume: 11
ISBN-10: 3-03719-081-7
ISBN-13: 978-3-03719-081-4
List Price: US$64
Member Price: US$51.20
Order Code: EMSTEXT/11
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This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at ETH Zurich.

The first part centers around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale's theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum.

The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed.

There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in dynamical systems.

Table of Contents

  • 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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