Symmetry in Mechanics: A Gentle, Modern Introduction 2004th Edition by Stephanie Frank Singer (PDF)

Description

“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.

User’s Reviews

Editorial Reviews: 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

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

⭐It’s a really great idea for an introduction, but at about the time when she introduces the covector, I got lost. It looked like a formal introduction of the concept and notation for no good reason. Perhaps more specific examples would help.

⭐I bought this book because I liked the idea of symmetry as a unifying principle in mechanics, treated in a book that I hoped would be easy to understand without major effort. The mathematics in this book was not unfamiliar to me and so I was surprised at how opaque the exposition was. The book is horribly written, with constant irrelevant asides and strange variations in the level of sophistication supposed of the reader. The exercises are often of the “guess what I’m thinking” type and often distract from the points she seems to be (trying) to make. Whatever the price offered for this book you must also figure in the time wasted. Don’t buy it, my advice.

⭐very needed book for specialists

⭐The author attempts to introduce the geometric view of mechanics with an intuitive, physically based presentation. That’s a worthy goal, but I’d rate the book as only a partial success. Singer nicely simplifies the presentation by limiting discussion of differential forms to two-forms and by confining the abstract definition of a differentiable manifold to an optional chapter; that optional chapter is quite good at building on simple examples to show why all the mathematical overhead is needed. The introduction to group theory and Lie groups is nice but could use more examples. In general, the intuitive presentation sometimes leaves nothing concrete to fall back on when a step in an argument is hard to follow.One complaint is that many of the notational choices are strange and will be confusing to physics students. For instance, a phase point is (r,p) instead of the more common (q,p); I often caught myself thinking incorrectly of Singer’s “r” as a radial coordinate rather than a Cartesian coordinate.Would I recommend this book to students who want to make the transition between the introductory Marion & Thornton or Goldstein level to the level needed for research in mechanics? Yes, but with the reservation that it should be used only as a companion to a more traditional presentation such as Arnold or Rasband. I don’t know how well I’d have learned this material had Singer been my textbook. I didn’t find it very easy to learn from Arnold’s book either; both together would have been a good combination.

⭐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.

⭐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.

⭐This is one of the very few books which I returned for refund. The subject is intrinsicly interesting, and there is a need for a serious introductory text addressing the subject of geometry and physics. This one badly falls short, – carelessly written, with numerous irrelevant asides. She seems even to fail to realize that there exist three distinct geometric solutions to the Kepler problem. The bound, elliptic case is only one. This book has supposedly been written for high-undergraduate students or early-year graduate students. It serves neither adequately.

⭐Hamilton力学の数学的理論を学ぼうとするとArnold，Abraham-Marsden，Marsden-Ratiu等の著作に頼ることになるが、数学を専門にしない人にとってはこれらの本は敷居が高すぎる。このSinger（女性！）の本は、理工系の大学学部で学ぶような初等的な数学（多変数微積分・ベクトル解析）の予備知識で読むことができ、懇切丁寧な説明で華麗なHamilton力学の理論を垣間見ることができる（私が知る）唯一の本。但し数学的知識（微分幾何学・Lie群論）が十分ある人には説明が冗長と感じるかも。

⭐

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