European Mathematical Society  m. audin
https://euromathsoc.eu/author/maudin
en

Hamiltonian Systems and Their Integrability
https://euromathsoc.eu/review/hamiltoniansystemsandtheirintegrability
<div class="field fieldnamefieldreviewreview fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><div class="tex2jax"><p>This book serves as an introduction to mathematical aspects of integrable Hamiltonian systems. In the first chapter, the author introduces symplectic manifolds, Poisson brackets on a symplectic manifold, Hamiltonians and integrable Hamiltonian systems (together with basic examples), as well as the classical Darboux theorem and its proof. The second chapter contains a description of symplectic action and the local ArnoldLiouville theorem on actionangle variables, as well as a global version of the theorem. In the third chapter, a relationship of integrability and the differential Galois group is treated. The author proves a strictly stronger version of the Zeglin lemma and the MoralesRamis theorem, giving a necessary condition for integrability in terms of the Galois group. In the fourth chapter, the author explains a relationship of the ArnoldLiouville theorem and algebraic curves via the Laxpair approach. The last two chapters are appendices containing basics of differential Galois theory and algebraic geometry (algebraic curves and the RiemannRoch theorem), which are used in the two previous chapters. The book is a nice introduction to integrability of Hamiltonian systems connecting it with two mentioned branches of mathematics, which were developed independently but which are deeply related to the studied topic. The book contains many nice illustrative examples (including the simple and spherical pendulum, the EulerPoinsot rigid body, the Lagrange and the Kowalevski top, harmonic and anharmonic oscillators and the HÃ©nonHeiles system). It also contains many exercises, the solutions of which help the reader to understand the theory. The book offers a nice, short and concise introduction to this attractive topic for mathematicians as well as physicists. For the latter, it also represents a natural opportunity to learn more not only about integrable systems but also about the basics of differential Galois theory and algebraic curves.</p>
</div></div></div></div><div class="field fieldnamefieldreviewreviewer fieldtypetext fieldlabelinline clearfix"><div class="fieldlabel">Reviewer: </div><div class="fielditems"><div class="fielditem even">skr</div></div></div><span class="vocabulary field fieldnamefieldreviewauthor fieldtypetaxonomytermreference fieldlabelinline clearfix"><h2 class="fieldlabel">Author: </h2><ul class="vocabularylist"><li class="vocabularylinks fielditem even"><a href="/author/maudin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">m. audin</a></li></ul></span><span class="vocabulary field fieldnamefieldreviewpublisher fieldtypetaxonomytermreference fieldlabelinline clearfix"><h2 class="fieldlabel">Publisher: </h2><ul class="vocabularylist"><li class="vocabularylinks fielditem even"><a href="/publisher/americanmathematicalsociety" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society</a></li></ul></span><div class="field fieldnamefieldreviewpub fieldtypenumberinteger fieldlabelinline clearfix"><div class="fieldlabel">Published: </div><div class="fielditems"><div class="fielditem even">2008</div></div></div><div class="field fieldnamefieldreviewisbn fieldtypetext fieldlabelinline clearfix"><div class="fieldlabel">ISBN: </div><div class="fielditems"><div class="fielditem even">9780821844137</div></div></div><div class="field fieldnamefieldreviewprice fieldtypetext fieldlabelinline clearfix"><div class="fieldlabel">Price: </div><div class="fielditems"><div class="fielditem even">USD 55</div></div></div><span class="vocabulary field fieldnamefieldreviewmsc fieldtypetaxonomytermreference fieldlabelhidden"><ul class="vocabularylist"><li class="vocabularylinks fielditem even"><a href="/msc/37dynamicalsystemsandergodictheory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37 Dynamical systems and ergodic theory</a></li></ul></span>
Mon, 06 Jun 2011 20:06:29 +0000
Anonymous
39339 at https://euromathsoc.eu