European Mathematical Society - sergio benenti
https://euro-math-soc.eu/author/sergio-benenti
enHAMILTONIAN STRUCTURES AND GENERATING FAMILIES
https://euro-math-soc.eu/review/hamiltonian-structures-and-generating-families
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Among the brilliant developments achieved by Differential Geometry along the XX century, the geometric formulation of Classical Mechanics has proved to be one of the most relevant. In this context, the Hamiltonian formulation has attired much attention of both mathematicians and physicists. For instance, from a mathematical point of view, symplectic Geometry finds essential inspiration in Hamiltonian Mechanics whereas physicists formalize many plausible models by means of the symplectic tools studied in Geometry.<br />
Given a manifold equipped with a symplectic structure, the submanifolds that are isotropic and coisotropic with respect to the symplectic structure are called Lagrangian. Many important concepts and definitions (or, according to many specialists, all the concepts and definitions!) in symplectic Geometry or in Hamiltonian Mechanics can be written in terms of these special submanifolds. Furthermore, if the manifold is a cotangent bundle with the canonical symplectic structure, a Lagrangian submanifold is locally generated by a function. This construction can be generalized by the so-called generating families to describe more general Lagrangian sets that are connected with some physical meaningful phenomena. The main topic of this book covers the definition, description and applications of generating families. For this purpose, the introduction of the notion of symplectic relation plays an important role.<br />
This reference is an enhanced version of a previous book edited in Russian. This new work gives an improved presentation of the theoretical part and deeper developments of its applications. In fact, the applications of the Lagrangian submanifolds and symplectic relations given in chapters 6, 7 and 8 are especially motivating. They present the Hamilton-Jacobi theory in geometric Optics, Hamiltonian Optics in Euclidean space and control of thermostatic systems respectively. There are many other scenarios where Hamiltonian systems play a key role. A comprehensive study of them would simply overwhelm the length of a single book. From this point of view, the choice done by this book could have been different, though the importance and elegance of applications in these chapters need not further justification.<br />
This book is aimed at both undergraduate and graduate students with just some initial knowledge in Algebra and Geometry. For this reason, the chapters try to include enough preliminaries to provide a gentle introduction to the topics covered by them. Specialists will also find a nice reference in this book specially, I think, with respect to the classical applications to Optics and thermostatics.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Marco CASTRILLON LOPEZ</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid, Spain</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book gives the basic definitions and results as well as some physical instances where the theory is applied, of Hamiltonian systems in terms of Lagrangian submanifolds and their generating families.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/sergio-benenti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">sergio benenti</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2011</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4614-1498-8</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/70-mechanics-particles-and-systems" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70 Mechanics of particles and systems</a></li></ul></span>Wed, 01 May 2013 16:57:08 +0000Anonymous45507 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/hamiltonian-structures-and-generating-families#comments