Spivak's Physics for Mathematicians: Mechanics

[ibkev]

Is it possible to briefly indicate what audiences Morin, Taylor, Spivak, and Arnold are intended for?

 

[S.G. Janssens]

I am very surprised! I think Arnold's writing is pretty confusing and sloppy. But I didn't think that somebody with your abilities would struggle with it! I always thought Arnold was written with people like you in mind.
If you don't mind, what about Arnold's book weren't you able to grasp? And did you know differential geometry/manifold theory before attempting Arnold?

Thank you for your confidence in my abilities. One of my advisers (a Russian) always raved about the book, but I was very much put off by precisely the sloppiness that you mention. When I attempted the read, I did know differential geometry. However, maybe I lacked (and lack) the maturity to keep the big picture in sight even when certain details are confusing or neglected. Perhaps I should give it another chance at some point.

 

[slider142]

I found Spivak's Physics for Mathematicians very clear. However, it is not a book on physics: it is really a book for people who are curious as to whether classical mechanics can be put on rigorous mathematical footing without the hand-waving and unstated assumptions that one frequently encounters in books on classical mechanics written for physicists. His use of the lever as a parable for this type of instruction is clever, and helps set up the type of reader he is aiming for: someone who has studied classical mechanics the usual way already, but is troubled by its ragged edges. Usually, the rigor comes in with Lagrangian mechanics, but then the rigor comes from a different direction (functional analysis), and the type of rigor needed often leaves students confused as to what types of physics the Lagrangian methodology is supposed to apply to, since previous mechanical instruction left the domain of applicability of certain things unclear. Spivak starts with this level of rigor from the very beginning, which makes the development very slow, but leaves all assumptions clear and unambiguous. It is, however, not suitable for a first text on mechanics. Possibly as a companion text, though, or a second text.

 

[smodak]

I found Spivak's Physics for Mathematicians very clear. However, it is not a book on physics: it is really a book for people who are curious as to whether classical mechanics can be put on rigorous mathematical footing without the hand-waving and unstated assumptions that one frequently encounters in books on classical mechanics written for physicists. His use of the lever as a parable for this type of instruction is clever, and helps set up the type of reader he is aiming for: someone who has studied classical mechanics the usual way already, but is troubled by its ragged edges. Usually, the rigor comes in with Lagrangian mechanics, but then the rigor comes from a different direction (functional analysis), and the type of rigor needed often leaves students confused as to what types of physics the Lagrangian methodology is supposed to apply to, since previous mechanical instruction left the domain of applicability of certain things unclear. Spivak starts with this level of rigor from the very beginning, which makes the development very slow, but leaves all assumptions clear and unambiguous. It is, however, not suitable for a first text on mechanics. Possibly as a companion text, though, or a second text.

I think this description is quite accurate. It is also a great companion to Chandrashekhar's Principia

 

[Joker93]

I found Spivak's Physics for Mathematicians very clear. However, it is not a book on physics: it is really a book for people who are curious as to whether classical mechanics can be put on rigorous mathematical footing without the hand-waving and unstated assumptions that one frequently encounters in books on classical mechanics written for physicists. His use of the lever as a parable for this type of instruction is clever, and helps set up the type of reader he is aiming for: someone who has studied classical mechanics the usual way already, but is troubled by its ragged edges. Usually, the rigor comes in with Lagrangian mechanics, but then the rigor comes from a different direction (functional analysis), and the type of rigor needed often leaves students confused as to what types of physics the Lagrangian methodology is supposed to apply to, since previous mechanical instruction left the domain of applicability of certain things unclear. Spivak starts with this level of rigor from the very beginning, which makes the development very slow, but leaves all assumptions clear and unambiguous. It is, however, not suitable for a first text on mechanics. Possibly as a companion text, though, or a second text.

Does it contain a lot of differential geometry? Will it suit somebody who wants to study the differential geometry aspect of mechanics while taking his first course at it(the course is being taught the usual physics way)? Also, does the rigor replace the intuition needed in order to understand the mathematics behind the physics or did the author provide the reader with both rigor and intuition?
Thanks!

 

[vanhees71]

I am very surprised! I think Arnold's writing is pretty confusing and sloppy. But I didn't think that somebody with your abilities would struggle with it! I always thought Arnold was written with people like you in mind.
If you don't mind, what about Arnold's book weren't you able to grasp? And did you know differential geometry/manifold theory before attempting Arnold?

Uups?!?? If Arnold's book is sloppy, what do you think about the usual physics books then? I thought Arnold is the perfect balance between math rigor and writing in a comprehensible way. Sometimes math is written in so formal a way that I, as a mathematically very interested theoretical physicist, can't make it far, because it's so unlively that I don't get any intuition. Of course, math must be rigorous, because otherwise, it's no math, but a textbook should also convey the intuitive side at list a bit, and there I thought Arnold is one of the few modern textbook writers who provide both sides. My counter example is Dieudonne, whose analysis textbooks are a nightmare although for sure mathematically at top level.

 

[S.G. Janssens]

Uups?!?? If Arnold's book is sloppy, what do you think about the usual physics books then?

Sorry for being imprudent: I know you did not ask me the question, but I would like to comment that I often find physics (text)books pretty hard to read, because they are usually not written in the strict theorem-proof style that suits me. Nevertheless, I keep making an effort because I think that learning physics merely from the (applied) mathematics literature would make me miss out on lots of interesting work done by the physicists (and theoretically inclined engineers) themselves.

 

[vanhees71]

For me it's the opposite. I think the entire fun of math is gone in this boring theorem-proof style. That's a cultural thing. Math books weren't written always in this way. There are the famous books by Courant and Hilbert "Mathematische Methoden der Physik" or the series on analysis by Smirnov which are not written in this Bourbaki style. I guess this style is more suitable for original research papers than textbooks.

 

[slider142]

It only introduces differential geometry concepts when they are needed, right before Lagrangian mechanics is explored. So the latter half of the text is best explored after you have familiarized yourself with the language of differential geometry.
As for intuition, it may help a little, but intuition isn't really the domain of this particular text. Rather, it explores the details of what we must assume in order for the mathematical models of physical phenomena that we are used to to be viable, and the precise places where they are no longer viable, so that we can base our intuition on a solid background of non-contradiction, known curious behaviors, and a knowledge of the disconnect between rigorous mathematical models and physical reality.

 

[Mark Harder]

Do you think that these books-which are intended for mathematics students-would be helpful for me during a first course in Classical Mechanics?

While the history of physics is interesting, I don't think learning it will help you in your course. For one thing, 'modern' approaches to classical mechanics have gone way beyond Newton, and you need to learn these to understand how theoretical mechanics is done (assuming that's the goal of taking the course)

 

[Mark Harder]

Did you ask the course instructor for recommendations?