Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems (Theoretical and Mathematical Physics) 2013th Edition by Gerd Rudolph (PDF)

Description

Starting from an undergraduate level, this book systematically develops the basics of• Calculus on manifolds, vector bundles, vector fields and differential forms,• Lie groups and Lie group actions,• Linear symplectic algebra and symplectic geometry,• Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory.The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics.The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.

User’s Reviews

Editorial Reviews: Review From the reviews:“The book is the first of two volumes on differential geometry and mathematical physics. The present volume deals with manifolds, Lie groups, symplectic geometry, Hamiltonian systems and Hamilton-Jacobi theory. … There are several examples and exercises scattered throughout the book. The presentation of material is well organized and clear. The reading of the book gives real satisfaction and pleasure since it reveals deep interrelations between pure mathematics and theoretical physics.” (Tomasz Rybicki, Mathematical Reviews, October, 2013) From the Back Cover Starting from an undergraduate level, this book systematically develops the basics of• Calculus on manifolds, vector bundles, vector fields and differential forms,• Lie groups and Lie group actions,• Linear symplectic algebra and symplectic geometry,• Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory.The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics.The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Gerd Rudolph and Matthias Schmidt have produced a brilliant two-volume contribution to the rapidly growing genre of applications of differential topology and differential geometry to mathematical physics. The material in these books emerged from courses the authors taught at the University of Leipzig over many years; in my experience, this type of class-tested material is usually far more organized and reader-friendly. I have found both volumes to be so clearly written that they can be used for independent study.There is always the difficult question of prerequisites. I would identify the absolute minimal mathematical prerequisites as linear algebra, point-set topology, an introductory course on Riemannian geometry, and a solid course in multivariable calculus (at the post-linear algebra level). The more undergraduate physics one has seen, the more physical connections one will understand. In particular, some exposure to classical mechanics, thermodynamics, and electromagnetism would be extremely helpful.Rudolph and Schmidt’s work is quite ambitious: Volume I contains 759 pages, and Volume II contains 830 pages. Topology and geometry have become so widely used in modern theoretical physics that most books in this genre typically focus on at most a handful of areas of application. While Rudolph and Schmidt discuss several applications, I believe that most readers will be drawn to these books primarily for their self-contained and beautifully structured treatment of gauge field theories. In my opinion, the study of connections on fiber bundles and their applications to gauge field theories represent the dominant themes of these books.Amazon’s “Look Inside” feature allows the potential buyer to read the Table of Contents, so there is no need for a lengthy discussion of contents on my part. Let me instead discuss some aspects of the books that are not immediately apparent from a cursory examination.The first observation should be preceded by the admission that I am a mathematician who works in mathematical physics; but I identify myself as a mathematician, both by training and disposition. As such, I am quite aware of the (often-discussed) language barrier that separates mathematicians from physicists. I have placed many reviews on Amazon identifying books on physics that are written in a manner that is accessible to mathematicians. To all my mathematical colleagues, I offer strong assurances that the two volumes under review are written with exceptional mathematical clarity and rigor. Indeed, they read like graduate mathematics texts (and I sincerely hope that Drs. Rudolph and Schmidt are not offended by that remark, for it is certainly intended as a compliment).The second observation is that these books are remarkably self-contained. The first six chapters of Volume I provide thorough introductions to smooth manifolds, vector fields, vector bundles, differential forms, and Lie groups. The first two chapters of Volume II then discuss principal fiber bundles and connections defined on them, relying heavily on the first six chapters of Volume I. This “highbrow” approach to connections, pioneered by Cartan, Ehresmann, and others, is central to the gauge field theory that occupies much of Volume II. Rudolph and Schmidt’s discussion of connections on fiber bundles is among the clearest I have ever encountered, not only because it is so carefully written, but also because it is entirely self-contained and does not send the reader off in search of other references for prerequisite material. How convenient for the student of this difficult, eclectic field to have all this material developed coherently and laid out for him in two books. Readers who have struggled to read the discussion of connections on fiber bundles in the classic books by Kobayashi and Nomizu will find the presentation in Rudolph and Schmidt far more reader-friendly. All of the difficult prerequisite material that Kobayashi and Nomizu presume of the reader is contained in Volume I of Rudolph and Schmidt.There are numerous other applications discussed in these books, including symplectic geometry, Hamiltonian systems and Hamilton-Jacobi Theory, spin structures, and the Atiyah-Singer Index Theorem, among others. The Introductions are short master classes in the complex and interrelated histories of the physical ideas and the mathematical notions that express them. Simply by reading the Introduction in each book, the reader can quickly determine that the authors are in complete command of this material, and that they also possess a deep understanding of its history. At the end of each chapter, the authors provide numerous helpful recommendations for original sources and additional reading: Volume I contains 318 references, while Volume II contains no less than 699. There are exercises scattered throughout both volumes (but no answers given).I recommend these remarkable books in the strongest possible terms to every serious student of gauge field theory, or more generally, to every student of applications of differential geometry to theoretical physics. The only references I am unaware of that provide comparable mathematical background along with the physical applications are the two volumes “Topology, Geometry, and Gauge Fields I and II” By Gregory L. Naber. If you decide that you have the necessary maturity and background to study Rudolph and Schmidt’s books, then I urge you to buy the two of them simultaneously. One need not read through all of Volume I before turning to Volume II; I found myself frequently moving back and forth between the two volumes as I initially studied them. I only wish that these masterful books had been available many years ago when I was trying to teach myself about gauge theories by poring over dozens of different books.

⭐This is a thorough summary of the subjects in the title. As the back cover says, it is written in thestyle of a mathematics text, that is, lots of proofs, examples, and exercises. Of course there are textswhich would get the physicist up to speed in a less complete coverage. But this is a book to work throughand keep as a reference when it comes to the fine details, all the conditions and restrictions that hold whenapplying the methods. Moreover, there is a generous supply of footnotes which point to easily accessible references for still more details and depth. I look forward eagerly to the next volume.

⭐Il libro è scritto bene e mostra i legami fra la geometria e la meccanica.

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