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Lectures on Dynamical Systems = Hami...

Zehnder, Eduard,

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Lectures on Dynamical Systems = Hamiltonian Vector Fields and Symplectic Capacities /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Lectures on Dynamical Systems/ Eduard Zehnder
Reminder of title:	Hamiltonian Vector Fields and Symplectic Capacities /
Author:	Zehnder, Eduard,
Published:	Zuerich, Switzerland :European Mathematical Society Publishing House, : 2010,
Description:	1 online resource (363 pages)
Subject:	Calculus of variations - Congresses. -
Online resource:	https://doi.org/10.4171/081
Online resource:	https://www.ems-ph.org/img/books/zehnder_mini.jpg
ISBN:	9783037195819

Lectures on Dynamical Systems = Hamiltonian Vector Fields and Symplectic Capacities /
Zehnder, Eduard,

Lectures on Dynamical SystemsHamiltonian Vector Fields and Symplectic Capacities /[electronic resource] :Eduard Zehnder - Zuerich, Switzerland :European Mathematical Society Publishing House,2010 - 1 online resource (363 pages) - EMS Textbooks in Mathematics (ETB).

Restricted to subscribers:https://www.ems-ph.org/ebooks.php

This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at the ETH Zurich. The first part centres 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.

ISBN: 9783037195819

Standard No.: 10.4171/081doiSubjects--Topical Terms:

1325460
Calculus of variations
--Congresses.

Lectures on Dynamical Systems = Hamiltonian Vector Fields and Symplectic Capacities /
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520 $a This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at the ETH Zurich. The first part centres 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.
650 0 7 $a Calculus of variations $v Congresses. $3 1325460
650 0 7 $a Dynamical systems and ergodic theory $2 msc $3 3480821
650 0 7 $a Ordinary differential equations $2 msc $3 3480850
650 0 7 $a Differential geometry $2 msc $3 3480811
650 0 7 $a Mechanics of particles and systems $2 msc $3 3480877
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