Spivak's Physics for Mathematicians: Mechanics

[Joker93]

Hello,
I will be enrolling in an undergraduate Classical Mechanics course and I was wondering if the book by Spivak "Physics for Mathematicians: Mechanics" would help me understand the concepts more in depth than usual.
Until the time that I will be taking the course, I will already have finished undergraduate course in General Relativity and Theoretical physics(separately). So, I think that I will have some knowledge of some of the concepts that are presented in this book; my background in mathematics will be a little bit more advanced than the rest of the students that will be taking the course.
Thanks in advance.

P.S. If anybody has used it or read it a little bit, what are your opinions?

 

[robphy]

Here's a little discussion of it at
http://mathoverflow.net/questions/5...-mathematicians-who-dont-know-the-first-thing ,
which points to a set of lecture notes http://alpha.math.uga.edu/~shifrin/Spivak_physics.pdf which preceded the book.

 

[Joker93]

Here's a little discussion of it at
http://mathoverflow.net/questions/5...-mathematicians-who-dont-know-the-first-thing ,
which points to a set of lecture notes http://alpha.math.uga.edu/~shifrin/Spivak_physics.pdf which preceded the book.

Thanks

 

[micromass]

I own the book and have read parts of it. I do not like it at all. It is very confusing, and a lot of it is unhelpful. For example, his first chapter discusses Newton's principia, which is notoriously difficult to understand. I don't understand why he takes so much time in discussing this book. Of course, if you're interested in the historical context, then this book might be helpful to you. Otherwise, I suggest you study classical mechanics from books written by physicists. Even for mathematicians, there are a lot of good alternatives such as Arnold or Marsden.

 

[Joker93]

I own the book and have read parts of it. I do not like it at all. It is very confusing, and a lot of it is unhelpful. For example, his first chapter discusses Newton's principia, which is notoriously difficult to understand. I don't understand why he takes so much time in discussing this book. Of course, if you're interested in the historical context, then this book might be helpful to you. Otherwise, I suggest you study classical mechanics from books written by physicists. Even for mathematicians, there are a lot of good alternatives such as Arnold or Marsden.

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

 

[micromass]

What do you mean with a first course in classical mechanics? How can you already have taken GR and theoretical physics without any knowledge of classical mechanics? What is the contents of the course?

 

[Joker93]

What do you mean with a first course in classical mechanics? How can you already have taken GR and theoretical physics without any knowledge of classical mechanics? What is the contents of the course?

Well, it's a complicated story, but I am self-studying everything that those courses have as prerequisites so I can successfully complete them.
The contents of the courses are the standard material that every university that teaches their undergraduates general relativity and theoretical physics contain.

 

[micromass]

Maybe it would be more helpful if you gave a list of things in classical mechanics you know (for example, you might already know kinematics and Newton's law), and the content of the course you'll take.

 

[Joker93]

Maybe it would be more helpful if you gave a list of things in classical mechanics you know (for example, you might already know kinematics and Newton's law), and the content of the course you'll take.

Well, I will be self-studying some part before I take the aforementioned courses(I have not taken any of them yet). I will self-study Lagrangian Mechanics(from Morin's book). Other than that, I just know about Newtonian Mechanics, but nothing fancier than first-year undergrad material.
The Classical Mechanics course's contents are(from a rough translation of the syllabus):
-Inertial reference frames and generalized coordinates
-Newtonian Mechanics
-Linear and non-linear oscillations
-Lagrangian formalism
-Calculus of variations
-Central potentials
-Gravity fields
-Conservation laws
-Oscillations of small magnitude
-Mechanics of rigid bodies
-Hamiltonian formalism
-Chaos
-Noether's theorem and symmetries

 

[micromass]

OK, thank you. In that case, you should take a look at Taylor and Gregory. These two books give an excellent coverage of the contents of your course. I have a slight preference over Taylor.
https://www.amazon.com/dp/189138922X/?tag=pfamazon01-20
https://www.amazon.com/dp/0521534097/?tag=pfamazon01-20

 

[Joker93]

OK, thank you. In that case, you should take a look at Taylor and Gregory. These two books give an excellent coverage of the contents of your course. I have a slight preference over Taylor.
https://www.amazon.com/dp/189138922X/?tag=pfamazon01-20
https://www.amazon.com/dp/0521534097/?tag=pfamazon01-20

Yeah, Taylor is great, I just borrowed it from the university library. But, as I will already have some knowledge of higher mathematics, won't the book by Spivak(or Marsden's or Arnold's book on Classical Mechanics) help me go deeper into the subject?

 

[Joker93]

@micromass Also, how can I PM you about something relating self-studying of mathematics?

 

[micromass]

Sure, but books like Taylor have a very different goal and scope than Arnold or Marsden.

 

[Joker93]

Sure, but books like Taylor have a very different goal and scope than Arnold or Marsden.

If I use Taylor and supplement it with one of those books so as to gain a deeper understanding of the differential geometry that's behind classical mechanics?

 

[micromass]

If I use Taylor and supplement it with one of those books so as to gain a deeper understanding of the differential geometry that's behind classical mechanics?

That would be an excellent thing to do!
Feel free to PM me any time!

 

[Joker93]

That would be an excellent thing to do!
Feel free to PM me any time!

The funny thing is that I can't find how to PM a user!

 

[micromass]

Click on username and then "start a conversation"

 

[Joker93]

@micromass One last thing: Which of these books would you suggest for me to supplement Taylor's?
1) Marsden's
2) Arnold's
3) Spivak
4) Other

 

[micromass]

@micromass One last thing: Which of these books would you suggest for me to supplement Taylor's?
1) Marsden's
2) Arnold's
3) Spivak
4) Other

I would go for Marsden. Spivak is very confusing and I don't like his treatment. Arnold might be too advanced however and while Arnold does treat differential geometry, if you haven't seen it before, then Arnold's treatment is not enough.

 

[ibkev]

Aren't Morin and Taylor roughly equivalent? (With the possible exception that Morin has a large number of problems with solutions.)

 

[micromass]

Aren't Morin and Taylor roughly equivalent? (With the possible exception that Morin has a large number of problems with solutions.)

Not really. The books are very different. They are meant for very different audiences too.

 

[Joker93]

I think that using both of them will be ideal!

 

[dextercioby]

I was a little curious to see how another book on physics could be written by a mathematician. [Mathematicians started writing serious books on physics in 1918, with the famous "Raum, Zeit, Materie" by Hermann Weyl and have been doing that quite a lot ever since. It's worth mentioning the 2nd (from a chronological perspective) cornerstone of the literature on Quantum Mechanics, the "Mathematische Grundlagen der Quantenmechanik" by John von Neumann (1932)].

Actually, it was not a bad writing by Spivak, au contraire. A little under 700 pages is a lot of material for the reader/student, but remarkable books take many pages to write, no doubt (incidentally, the only exception I could mention is also in the field of classical mechanics, the gem by Lev Landau and Evghenii Lifschitz). The source of inspiration for the author appears to me to be given by classical (i.e. pre-1950) books, of which I mention the book by W.H. Osgood ("Classical Mechnics", 1st Ed. 1937). This book, though mathematical in nature, has a double advantage compared to, let's say, Goldstein or Morin+Taylor or Marion+Thornton:
- Puts emphasis on the long history of the topic, starts off with a careful (perhaps boring) analysis of Newton's thinking. Never leaves historical notes, sending the reader to the bibliographical items containing original material from the golden years of CM in the 19th century.
- Is mathematically accurate and balances this rigor with the descriptive style needed by a physics book, especially in mechanics.

Spivak's work is necessary, because it's the missing step in the overall literature on this important topic between the frightening (to me, at least) books by Arnold and Abraham+Marsden and the plethora of purely physics books of which I mentioned 4 in the previous paragraph. The only real question is: how relevant is the study of this book for someone wanting to go all the way in physics? This is judged only by the real interest of the targeted reader: aiming to become a mathematical physicist, for which I believe it to be helpful. Using the methods of differential geometry in the study of classical mechanics - gently presented in this book - prepares the reader for the thorough study of General Relativity under the guidance of R. Wald and Hawking+Ellis, or prepares him for a deep understanding of gauge field theory.

 

[Joker93]

I was a little curious to see how another book on physics could be written by a mathematician. [Mathematicians started writing serious books on physics in 1918, with the famous "Raum, Zeit, Materie" by Hermann Weyl and have been doing that quite a lot ever since. It's worth mentioning the 2nd (from a chronological perspective) cornerstone of the literature on Quantum Mechanics, the "Mathematische Grundlagen der Quantenmechanik" by John von Neumann (1932)].

Actually, it was not a bad writing by Spivak, au contraire. A little under 700 pages is a lot of material for the reader/student, but remarkable books take many pages to write, no doubt (incidentally, the only exception I could mention is also in the field of classical mechanics, the gem by Lev Landau and Evghenii Lifschitz). The source of inspiration for the author appears to me to be given by classical (i.e. pre-1950) books, of which I mention the book by W.H. Osgood ("Classical Mechnics", 1st Ed. 1937). This book, though mathematical in nature, has a double advantage compared to, let's say, Goldstein or Morin+Taylor or Marion+Thornton:
- Puts emphasis on the long history of the topic, starts off with a careful (perhaps boring) analysis of Newton's thinking. Never leaves historical notes, sending the reader to the bibliographical items containing original material from the golden years of CM in the 19th century.
- Is mathematically accurate and balances this rigor with the descriptive style needed by a physics book, especially in mechanics.

Spivak's work is necessary, because it's the missing step in the overall literature on this important topic between the frightening (to me, at least) books by Arnold and Abraham+Marsden and the plethora of purely physics books of which I mentioned 4 in the previous paragraph. The only real question is: how relevant is the study of this book for someone wanting to go all the way in physics? This is judged only by the real interest of the targeted reader: aiming to become a mathematical physicist, for which I believe it to be helpful. Using the methods of differential geometry in the study of classical mechanics - gently presented in this book - prepares the reader for the thorough study of General Relativity under the guidance of R. Wald and Hawking+Ellis, or prepares him for a deep understanding of gauge field theory.

Well, I want to become a theoretical physicist, so I think that books like this can only do some good.
Did you read the book? Is it well-written?

 

[dextercioby]

Micromass criticized it, but I liked what I read, including the historical context. I couldn't have gone through all of it for the lack of time, of course. I recommend it as an alternative to a classical physics one,

 

[Joker93]

Micromass criticized it, but I liked what I read, including the historical context. I couldn't have gone through all of it for the lack of time, of course. I recommend it as an alternative to a classical physics one,

Ok, thanks. I might read a part of it to see if it fits me

 

[vanhees71]

Well, the famous example of "Raum - Zeit - Materie" (i.e., "Space - Time - Matter") by Weyl and also von Neumann's book on the mathematical foundations of QT cement my prejudice that one should only read the math part of such books and leave the discovery of new physical models (or even theories) to the physicists. Weyl's attempt to unify electromagnetism and gravity by gauging the dilation symmetry of matter-free GR was already wrong in concept before he even could write it down since simply the geometric extension of objects doesn't depend on their "electromagnetic history", as was immediately pointed out by Einstein and in very harsh criticism by Pauli when Weyl published the idea. Otherwise it's a brillant book, and I find it provides a very intuitive picture of the meaning of the various formalities of tensor calculus. It can help to deepen the understanding of the theory of (pseudo-)Riemannian manifolds a lot. Similar things hold true for von Neumann's famous book. It's brillant in its foundations concerning the mathematics of unbound self-adjoint operators and the proper meaning of continuous spectra of such operators, i.e., in clarifying the math, but I don't like to comment on the physics/interpretational part...

I don't know Spivak's book. So I can't say anything about it. I know Arnold's book, and I find it brillant. I've not even have anything to complain about the physics side ;-)).

 

[S.G. Janssens]

I know Arnold's book, and I find it brillant. I've not even have anything to complain about the physics side ;-)).

I feel compelled to make a confession. While classical mechanics is my favorite part of physics (indeed, it is one of the few parts for which I feel I have some sort of understanding) and while I consider myself reasonably well-trained in mathematics, I have never been able to read Arnold's book.

Good to have that off my chest.

 

[micromass]

I feel compelled to make a confession. While classical mechanics is my favorite part of physics (indeed, it is one of the few parts for which I feel I have some sort of understanding) and while I consider myself reasonably well-trained in mathematics, I have never been able to read Arnold's book.

Good to have that off my chest.

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?

 

[smodak]

I really like the Spivak's mechanics book which I do not find confusing at all. But I guess everybody has a slightly different preference.

 

[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!