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Symmetry in Mechanics: A Gentle, Modern Introduction, by Stephanie Frank Singer
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"And what is the use," thought Alice, "of a book without pictures or conversations in it?" -Lewis Carroll This book is written for modem undergraduate students - not the ideal stu dents that mathematics professors wish for (and who occasionally grace our campuses), but the students like many the author has taught: talented but ap preciating review and reinforcement of past course work; willing to work hard, but demanding context and motivation for the mathematics they are learning. To suit this audience, the author eschews density of topics and efficiency of presentation in favor of a gentler tone, a coherent story, digressions on mathe maticians, physicists and their notations, simple examples worked out in detail, and reinforcement of the basics. Dense and efficient texts play a crucial role in the education of budding (and budded) mathematicians and physicists. This book does not presume to improve on the classics in that genre. Rather, it aims to provide those classics with a large new generation of appreciative readers. This text introduces some basic constructs of modern symplectic geometry in the context of an old celestial mechanics problem, the two-body problem. We present the derivation of Kepler's laws of planetary motion from Newton's laws of gravitation, first in the style of an undergraduate physics course, and x Preface then again in the language of symplectic geometry. No previous exposure to symplectic geometry is required: we introduce and illustrate all necessary con structs.
- Sales Rank: #1325776 in Books
- Brand: Brand: Birkhäuser
- Published on: 2013-10-04
- Released on: 2013-10-04
- Original language: English
- Number of items: 1
- Dimensions: 9.25" h x .47" w x 6.10" l, .81 pounds
- Binding: Paperback
- 193 pages
- Used Book in Good Condition
Review
"Symmetry in Mechanics is directed to students at the undergraduate level and beyond, and offers a lovely presentation of the subject . . . The first chapter presents a standard derivation of the equations for two-body planetary motion. Kepler's laws are then obtained and the rule of conservation laws is emphasized. . . . Singer uses this example from classical physics throughout the book as a vehicle for explaining the concepts of differential geometry and for illustrating their use. These ideas and techniques will allow the reader to understand advanced texts and research literature in which considerably more difficult problems are treated and solved by identical or related methods. The book contains 122 student exercises, many of which are solved in an appendix. The solutions, especially, are valuable for showing how a mathematician approaches and solves specific problems. Using this presentation, the book removes some of the language barriers that divide the worlds of mathematics and physics."
―Physics Today
"This is a very interesting book. Those educated in traditional mechanics will acquire [from reading it] knowledge of modern mathematics hidden beyond traditional concepts in the realm of celestial mechanics, [and] . . . pure mathematicians will understand how their discipline enters into practical problems. The author shows how fundamental concepts of symplectic geometry implicitly occur in mechanics . . . the mathematical presentation is ingenious and subtle. There are a lot of exercises for the reader and the solutions of most of them are given in a separate chapter. I can highly recommend this book to undergraduate and PhD students . . . it is ideally suited for teaching a course on the subject."
―Mathematical Reviews
Most helpful customer reviews
47 of 48 people found the following review helpful.
A welcome book
By A Customer
There are a number of books available on the "geometric" view of physics (Classical Mathematical Physics, by Thirring, The Geometry of Physics, by Frankel, and Foundations of Mechanics, by Abraham & Marsden). The size, level of sophistication and extensive background assumed by these books can be very intimidating. On the other hand, the subject "looks" beautiful, and the benefits of using geometric intuition are desirable to many people.
Singer's book stands class of its own in these respects. All the basics of the geometrical "machinery" are there, in a book that is only 224 pages in length. Chapter one starts with a standard derivation of the equations of the "two-body planetary motion" problem; subsequent chapters proceed to introduce the necessary modern geometrical and mathematical concepts (differential geometry). The final chapter then revisits the "planetary motion" problem using the modern concepts previously introduced. Excellent!
There are some misprints, but the author has a Web page of errata. The book has numerous exercises, with many solutions included. I find myself rereading parts of this book over and over.
Reader be warned; the concepts are new, and it does take work to internalize them. However, this is the most accessible book on the subject available, and also one of the most affordable. The author references many other books, for the reader who wants to go further in the mastery of this subject (one excellent book which is not mentioned, however, is "Differential Forms: A Complement to Vector Calculus", by Weintraub). Enjoy!
18 of 18 people found the following review helpful.
A nice introduction to modern methods
By LB
I think the previous review is a bit harsh, and that the book's intents are not what this reviewer expected. I don't think it was the author's intent to write a comprehensive treatise on the subject. The book simply aims at introducing undergraduate students to the use of symmetry in simplifying the analysis of classical mechanic problems, nothing more. If you want a comprehensive treatise, you probably want to read V.I. Arnols's "Mathematical methods in classical mechanics". If what you want is a simple introduction where all the steps are worked out in details, then this book is a good starting point, and I think this is what the author intended. At any rate, the cost ($$$) is quite reasonable.
9 of 9 people found the following review helpful.
Refreshingly different
By Cybertronian
There are two classes of books in mechanics: the extremely physical, which are intended to teach you how to solve problems but lack any mathematical rigour, and the mathematical ones, where the examples are generally one-line statements without any explanation. This book sits exactly in the middle of both: if you are a physicist (or mathematician for that matter) with a fair knowledge of classical mechanics and you understand the basics of Hamiltonian systems, but you want to expand your horizon with momentum maps and symplectic reduction, but you don't understand anything of the hardcore abstract books by mathematicians or you are afraid of them, this is where you should put your money.
Physicists usually simplify their equations by using symmetry in a rather ad hoc way; intuition tells you that a rotation around a certain axis does not change anything or that the system is invariant under translations, or that angular momentum is conserved in a certain direction. Symplectic reduction is the systematic study of these symmetries and how to simplify you equations with them. Don't expect to be shocked because most of the analyses can be carried out without knowing anything about symplectic reduction, but it can aid your life if you are working on more complicated systems, where your intuition does not help you very much (or if you just want to impress someone with your knowledge of mathematical mechanics).
The book does not go deeply into the material, but it explains the basics clearly (symplectic two-form, momentum maps, Lie derivative, reduction...) without being pedantically mathematical. Don't expect any proofs or general theorems; e.g. the author uses (dual) MATRIX Lie groups/algebras, which are intuitive for the physicist (just apply the matrices to your coordinate basis and that's it, quick and dirty) but not as general as the idea of coadjoint orbits of an abstract Lie algebra.
I have tried to go through the mathematics library on symplectic topology and symplectic reduction but have never come very far - and in the cases I thought I understood the concepts I found out that I could do absolutely nothing with it in practice, because I had never seen an actual calculation. After reading this book I must say that I have more confidence reading and understanding them. The book prepares you for more to come, which is exactly what it's aimed at. Instead of giving you the dry reality of modern mathematics wrapped in complete generality, it gives you the juicy extract of what it's all about, it lets you think about it, and use it in simple situations. If you want to go beyond this book, you'll have to have a firm knowledge of Lie groups, Lie algebras, and differential geometry, but for this book, you just need undergraduate physics and mathematics.
The book comes with lots of exercises and to some the answers are given at the back. It's a short and easy introduction to the uses of symmetry (reduction) in Hamiltonian mechanics, and it's good value for your money. I am happy to have it and I can only recommend it.
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