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Physics for Mathematicians, Mechanics I Hardcover – December 6, 2010
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About the Author
- Item Weight : 2.8 pounds
- Hardcover : 749 pages
- ISBN-10 : 0914098322
- ISBN-13 : 978-0914098324
- Publisher : Publish or Perish; first edition (December 6, 2010)
- Language: : English
- Customer Reviews:
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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 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.