Physics for Mathematicians, Mechanics I by Michael Spivak | (PDF) Free Download

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From the Preface: The purpose of this book, or possibly series of books, is indicated precisely by the title Physics for Mathematicians. It is only necessary for me to explain what I mean by a mathematician, and what I mean byphysics.By a mathematician I mean some one who has been trained in modern mathematics and been inculcated with its general outlook. … And by physics I mean — well, physics, what physicists mean by physics, i.e., the actual study of physical objects … (rather than the study of symplectic structures on cotangent bundles, for example). In addition to presenting the advanced physics, which mathematicians find so easy, I also want to explore the workings of elementary physics … which I have always found so hard to fathom.As these remarks probably reveal, basically I have written this work in order to learn the subject myself, in a form that I find comprehensible. And readers familiar with some of my previous books probably realize that this has pretty much been the reason for those works also. …Perhaps this travelogue of an innocent abroad in a very different field will also turn out to be a book that mathematicians will like.

User’s Reviews

Editorial Reviews: About the Author Michael Spivak is the author of Calculus and the 5 volume work Comprehensive Introduction to Differential Geometry.

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

⭐I give FIVE STARS to Spivak’s “Physics for Mathematicians, Mechanics I” even though I find important things wrong with it. If I could, I’d give this book ten stars because it feels as if it was written for me when it asks many of the same questions I wondered about and tries, mostly successfully, to address them. I knew immediately that I would buy this book when I saw that in the first few pages it addresses the “proof” of the law of lever as presented in Mach’s “The Science of Mechanics” and also when Cohen and Whitman’s new translation of the Principia is a prominent reference.This book is a worthy companion to Chandrasekhar’s “Newton’s Principia for the Common Reader” because it goes into wonderful detail in presenting Newton’s approach to (celestial) mechanics. I love the geometric proofs and Spivak goes so far as to take some of Newton’s complex diagrams apart and presents them a few steps at a time. Spivak’s is both simpler and more detailed than Chandrasekhar, and even avoids the way Chandrasekhar mucks up Newton’s clean, precise, and razor sharp proof that the gravitational force within a spherical shell is zero. Chapter 2, on Newton’s Analysis of Central forces, is wonderful (even with some of its flaws) and I will be using some of the results in a project I’ve undertaken concerning the gravitational field of thin disks.Now for what’s wrong with the book. These criticisms arise because of the approach I take as an Electrical Engineer (Control Theory); I’m not a physicist or mathematician.(1) This book is in dire need of a strong and determined editor; there is almost no consistency in the presentation. For example, in the first few pages of Chapter 2, some of the equations are labeled with numerals (1), (2); some with (*), (**), and (*’); and some alphabetically. No big deal, but it makes it hard to go back and find things.(2) Notation: Too loosey-goosey. As an example, in chapter 2 there is the phrase ” … and let c be any particle”. What attribute of the particle does c represent? It’s pretty clear that it is position (I think) (and, therefore, a vector), yet it is not indicated using bold face as is done for the velocity vector, v. An experienced reader can straighten these things out with careful reading, but the inconsistency is distracting and time consuming; especially since the book tries to be precise. (3) Differential Geometry: The author states outright and immediately that Differential Geometry is almost a prerequisite. Too bad. Fortunately, I’m able to go through the first few chapters, but wish I could do the really good stuff later in the book.In all, this is a very good book to study if you really want to know and understand fundamental questions in mechanics. And to appreciate Newton’s incredible Principia – as mind boggling today as 300+ years ago.

⭐My review will be mostly comparing the book to Spivak’s lecture notes. They are incredibly different.The first ~430 pages are dedicated to Newtonian mechanics (including central potential, rigid body motion, and fictitious forces). I’ve noticed most physics textbooks just give this as “God given”, but Spivak actually gives some intuition behind what’s going on. This book is the perfect foil for Morin’s “Introduction to Classical Mechanics”.The discussion of constraints is quite thorough. It begins with rigid body motion, and generalizes it in a beautiful way. Most other physics textbooks leave out any discussion of what to do with constraints (c.f. Arnold’s “Mathematical Methods of Classical Mechanics” or even Goldstein).Spivak discusses variational principles — not just the principle of stationary action, but others too. Euler’s equation derived from variation, Hamilton’s principle, Maupertuis’ principle of least action, Jacobi’s version of the principle of least action, and symmetry in variational calculus. There is a minor typo on page 466 (“Jacobi’s form of the principal [sic] of least action.”) and it is quite clear that differential geometry is assumed. (Well, Spivak suggests that the first two volumes of “A Comprehensive Introduction to Differential Geometry” should be read before hand.)There is a thorough discussion of Lagrangian and Hamiltonian mechanics from the differential geometric perspective. It’s not completely abstract, it’s amazingly grounded in physical intuition. There’s an entire chapter (26 pages) dedicated to the Hamilton-Jacobi theory.The only problem I have with the book is that classical field theory is not covered. Also gauge transformations are mentioned only once in passing. But this book is a wonderful introduction to mechanics for mathematicians, it will save a lot of frustration for mathematical physicists.

⭐This large book has the same spirit of the author’s book A Compreensive Introduction to Differential Geometry. And, as that one, is pretty uncommon. The premises of the book are great: to analyse, besides the advanced mathematical tools avaiable to theoretical Physics (tangent and cotangent bundles, sympletic geometry, etc), the common concepts of elementary Physics with minute details. It is perhaps unnecessary to point out that not many books on Physics do that nowadays. The study of inclined planes is symbolic of the spirit of the book. Spivak explains Archimedes argument, and later gives a complete description of the whole process using rigid body dynamics. The theoretical physicist perhaps never took the pains to do that, but the process should work in some way or another for the whole structure to be consistent. As for the subject, it covers essentially the whole subject of Classical Mechanics, from elementary portions to Lagrange’s and Hamilton’s equations. The book should interest not only mathematicians, contrary to Spivak’s opinion, but theoretical physicists as well, who want to have a well presented and connected account of the mathematical foundations of Mechanics. Is it possible to learn Mechanics from this work? I believe that some portions really could be used for that. Anyway, for someone who already understands Mechanics, is a pleasant fountain of knowledge of the Queen of physical Sciences, Classical Mechanics. And, as usual in Spivak’s books, a lot of historical notes illustrate how the subject evolved.I guess that this is a book which will attract more and more attention as the time passes, and eventually become a classic. Let’s just hope that Spivak completes his project of writing Physics books for Mathematicians.

⭐This book has a very specific audience: those with a pure Mathematics foundation who are interested in Physics. If you’ve always been frustrated by the lack of rigor and cohesion in Physics courses but are interested in the content this is the book for you. The same author wrote the best introduction to calculus there has ever been. He has a clear and concise way of writing, and takes you on a Journey from the foundations of classical physics through to some of the generalizations. I truly hope he writes the implied follow ups, but he is old and may not get to it. Still, despite the fact this book doesn’t cover everything, what it does cover, and the way it covers it, is worth the price. Those without rigorous training may find they prefer other resources.

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