03077nam a22003735a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001700168084003600185100003000221245013000251260008200381300003400463336002600497337002600523338003600549347002400585490003900609506006600648520169300714650003502407650004602442650004102488650003102529650004402560856003202604856006702636113-100519CH-001817-320100519234500.0a fot ||| 0|cr nn mmmmamaa100519e20100519sz fot ||| 0|eng d a978303719581970a10.4171/0812doi ach0018173 7aPBKQ2bicssc a37-xxa34-xxa53-xxa70-xx2msc1 aZehnder, Eduard,eauthor.10aLectures on Dynamical Systemsh[electronic resource] :bHamiltonian Vector Fields and Symplectic Capacities /cEduard Zehnder3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2010 a1 online resource (363 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Textbooks in Mathematics (ETB)1 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aThis 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.07aCalculus of variations2bicssc07aDynamical systems and ergodic theory2msc07aOrdinary differential equations2msc07aDifferential geometry2msc07aMechanics of particles and systems2msc40uhttps://doi.org/10.4171/081423cover imageuhttps://www.ems-ph.org/img/books/zehnder_mini.jpg