# Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities

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*Eduard Zehnder*

A publication of the European Mathematical Society

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.