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Physics for Mathematicians. Mechanics I. Hardcover

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Amazon.com: 5 reviews
58 of 59 people found the following review helpful
Best introduction to mechanics 7 May 2011
By A. Nelson - Published on Amazon.com
Format: Hardcover Verified Purchase
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.
60 of 62 people found the following review helpful
Fascinating account of Mechanics 27 Feb 2011
By Guilherme - Published on Amazon.com
Format: Hardcover Verified Purchase
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.
15 of 15 people found the following review helpful
Great Book with blemishes 18 Sep 2013
By MzF - Published on Amazon.com
Format: Hardcover Verified Purchase
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.
8 of 9 people found the following review helpful
It's great to find a "kindred spirit" 3 Jun 2013
By Steven N. Evans - Published on Amazon.com
Format: Hardcover Verified Purchase
Like the author, I'm a mathematician who had difficulties with the way that physics is often presented in a manner that makes it look like mathematics (or at least the Platonic ideal of mathematics), with conclusions deduced from axioms by inexorable logic, but when one examines the arguments more closely there are often all sorts of unstated assumptions that creep in. Spivak does a wonderful job of clarifying such basic issues as why it is that "rigid bodies" can be treated as Newtonian "particles" in appropriate circumstances and why combining forces using the usual rules of vector addition is not something that appears to be derivable from Newton's laws. These and similar matters confused me in the way I was taught physics, and it's great to find that a mathematician of Spivak's stature was similarly perplexed and see how he has sought to clear up that perplexity.
Five Stars 18 Aug 2014
By Thomas R. Schulte - Published on Amazon.com
Format: Hardcover Verified Purchase
Excellent book and excellent service!
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