Of course, everyone knows about levers...

October 4, 2014 3:38 PM

Elementary Mechanics from a Mathematician's Viewpoint [direct link to large PDF] by Michael Spivak - notes from his eight 2004 lectures (which eventually became a book). See the quote inside to get the flavor of it.

These lectures are based on a book that I am writing, or at least trying to write. For many years I have been saying that I would like to write a book (or series of books) called Physics for Mathematicians. Whenever I would tell people that, they would say, Oh good, you're going to explain quantum mechanics, or string theory, or something like that. And I would say, Well that would be nice, but I can't begin to do that now; first I have to learn elementary physics, so the first thing I will be writing will be Mechanics for Mathematicians.

So then people would say, Ah, so you're going to be writing about symplectic structures, or something of that sort. And I would have to say, No, I'm not trying to write a book about mathematics for mathematicians, I'm trying to write a book about physics for mathematicians; of course, symplectic structures will eventually make an appearance, but the problem is that I could easily understand symplectic structures, it's elementary mechanics that I don't understand.

Then people would look at me a little strangely, so I'd better explain what I mean. When I say that I don't understand elementary mechanics, I mean, for example, that I don't understand this:

           :......,                                             
           :......,                                             
           :......,                                             
           :......,                                             
           :......,                                             
           :......,                                    ;;;;     
           :......,                                    ,..,     
           :......,                                    ,..,     
   ##########################################################   
                      /\                                        
                     .  .                                       
                     ....                                       

Of course, everyone knows about levers. They are so familiar that most of us have forgotten how wonderful a lever is, how great a surprise it was when we first saw a small body balancing a much bigger one. Most of us also know the law of the lever, but this law is simply a quantitative statement of exactly how amazing the lever is, and doesn't give us a clue as to why it is true, how such a small force at one end can exert such a great force at the other.

posted by Wolfdog (24 comments total)

Thanks! I did a senior undergrad project on Differential Geometry as an honors math major and I loved the kooky illustrations on Spivak's Differential Geometry textbooks... "All the Way with Gauss-Bonet!"
posted by Schmucko at 3:55 PM on October 4, 2014

And hey, maybe there'll be the fifth section, on Lie groups. I'm charmed by the idea of starting with defining mass, and ending up...pretty far away from that.
posted by leahwrenn at 3:56 PM on October 4, 2014

This should be for me, but it isn't. Because it doesn't address the problem I had in Phys 1a / Mechanics — do I ignore the friction and write the equation for tension, or do I ignore the tension and write the equation for friction? Because goddamn if I didn't pick the wrong one every single time.

(But actually it does look interesting and I look forward to reading it. Thanks for the post.)
posted by benito.strauss at 4:09 PM on October 4, 2014

Hoping he'll eventually write a chapter on fuckin' magnets.
posted by ZenMasterThis at 4:14 PM on October 4, 2014 [5 favorites]

I'm sure that there's some idea here, but man do I hate Spivak's style. Took a course in college that used his Calculus on Manifolds and it was dreadful.

In any case, this series of lectures brings to mind Feynman's discussion on "why" questions. It seems like Spivak is asking (and answering) why questions that go pretty far. Whether or not this is terribly useful is arguable (as a mathematician I have never felt particularly at sea with mechanics), but as a thought experiment it is at least interesting.
posted by TypographicalError at 5:11 PM on October 4, 2014 [1 favorite]

Calculus on Manifolds played a strong role in sealing my decision to become a mathematician. That book gave me insights that served me far beyond what you would ever reasonably expect from a wee little book on multivariable calculus. And of course I didn't know at the time how it would pay off for me later, but I did know that it made me feel smart. Perhaps it was just a fortuitous match between the difficulty of the exercises and my own level of understanding at the time. But I can't think of any other book that ever gave so many OMIGOSH NOW I REALLY GET IT THAT IS SO AWESOME moments.
posted by Wolfdog at 5:19 PM on October 4, 2014 [4 favorites]

I'm sure that there's some idea here, but man do I hate Spivak's style. Took a course in college that used his Calculus on Manifolds and it was dreadful.

There are two kinds of mathematicians in this world... those who like CoM and those who don't. I don't. All of Spivak's books suffer from a very picky approach to proof from axioms. He's the sort of mathematician who would mark a proof wrong if you made grammatical mistakes, which leads him into some convoluted arguments (in order to convince himself of internal consistency) of his own which obscure rather than reveal. I think a sort of Poincare-like pragmatism gets you farther in Differential Geometry. But, his great american book on differential geometry is a great resource on lots of arguments you might not otherwise seen explained.

Hoping he'll eventually write a chapter on fuckin' magnets.

To understand fuckin' magnets you have to give up the Newtonian idea of 'Force.' So, I'm not sure he's going to do that in a book which seems to be about Newtonian mechanics. Although he seems interested in deconstructing 'rigid bodies' which is the first step.
posted by ennui.bz at 5:46 PM on October 4, 2014 [2 favorites]

Magnets require spin, and spin is something that is very convenient and functional mathematically but deeply weird and confusing physically, so I suspect a mathematician is not the right person to help there.
posted by kiltedtaco at 6:30 PM on October 4, 2014

There are two kinds of mathematicians in this world... those who like CoM and those who don't.

I found CoM intermittently enraging when I worked through it for a HS math class (yeah, I got shoved into my locker a lot). My copy is permanently warped from when I hurled it into a wall; I think that was at the part where he tries to make a cross product in R^2 into something reasonable and consistent. But when I went on to take Differential Geometry courses for my college major, I felt like the ludicrously finicky underpinnings made it a lot easier for me to attach that "Poincare-like pragmatism" on the top. Ultimately I'm glad I studied from it, but I'm not sure I'd do it again.
posted by dorque at 6:44 PM on October 4, 2014

It dawned on me as I was TAing calculus in grad school that all those integrals that appeared in AP Physics five years earlier with these hand-wavy non-intuitive explanations would have made perfect sense if only someone had said the words "Riemann sum". In other words, I am very much the target audience here.
posted by hoyland at 6:44 PM on October 4, 2014

Okay, well he lost me right near the beginning when he talks about "uniform gravity". I've measured the change in gravitation force just between a table top and the floor and there is nothing uniform about it. As for the change between the the ground and the top of the Tokyo Tower, the stretching of the spring is certainly not "exactly the same."

Richard Feynman was adamant about his criticism of sloppy physics, for example ramp and rolling ball experiments that ignored rotational inertia. I'm done with Spivak.
posted by JackFlash at 7:45 PM on October 4, 2014

Okay, well he lost me right near the beginning when he talks about "uniform gravity". ... Richard Feynman was adamant about his criticism of sloppy physics, for example ramp and rolling ball experiments that ignored rotational inertia. I'm done with Spivak.

You are missing the point, which is that locally gravity is more or less "uniform" (it certainly would have appeared to be to Newton), or at the very least the fluctuations are so minor as to have little to no impact, and that from this we can proceed to set up the rest of Newton's results. Spivak himself says "of course, [claiming uniformity is] not really true for the force of gravity, but it's true to a very good approximation for the sort of distances above the earth's surface that we are concerned with." This is in no way as major as ignoring rotational inertia.

If we say your floor is at sea level and your tabletop is 1 metre off the ground, then if we take the acceleration due to gravity at sea level to be 9.80665 ms^-2, acceleration due to gravity on the surface of your table is 9.806646931 ≈ 9.80665 ms^-2. That is, no change to 5 d.p. Yet you say there's nothing uniform about it? Sounds like you're just being snobbish.
posted by Quilford at 8:07 PM on October 4, 2014 [3 favorites]

Okay, well he lost me right near the beginning when he talks about "uniform gravity".

Be fair, he's quoting Newton there. But yeah, he lost me right about that same point, when he was meandering around Newton's principles and then beating them to death. I just don't see the point of any of this. This isn't "physics for mathematicians," it's "my strange beliefs about physics that allow me to demonstrate strange ideas about math's relevance to physics." Nope, this is sloppy physics and sloppy math.

Ulam once said, "the mathematicians know a great deal about very little and the physicists very little about a great deal." Spivak seems to know very little about a very few things. Unfortunately, those are the very things he tries to use as fundamentals.
posted by charlie don't surf at 8:13 PM on October 4, 2014 [1 favorite]

Nope, this is sloppy physics and sloppy math.

And that's sloppy rhetoric. Care to point out some actual instances of 'sloppy' physics or maths in the text?

Good grief. It's clear that the first lecture is designed to walk the reader through the Principia because that's where Newton went and established all his laws from first principles.

Spivak seems to know very little about a very few things.

Given that you were lost 10 pages into the 100 page book, I don't think you're particularly qualified to pass judgement on it.
posted by Quilford at 8:34 PM on October 4, 2014 [2 favorites]

Care to point out some actual instances of 'sloppy' physics or maths in the text?

Well, I have a passing familiarity with the English language and I think I know what "exactly the same" means. I also have a passing familiarity with physics and think I know what the term "r" means in Newton's law of gravitation.

So I have a hard time reconciling those two facts with "The next part of the experiment is one that probably no one except a tourist like myself might be willing to perform, taking the same measurement at the top of Tokyo Tower. Once again, it seems that the spring is stretched by exactly the same amount."

Because I have made similar measurements and literally seen with my own eyes that the spring is not stretched by the same amount, nor would I expect it to be.
posted by JackFlash at 12:01 AM on October 5, 2014

I followed the book far beyond the point of futility, all the way to the end. If you want an example of sloppy physics and math, I recommend the final example at the end of lecture 8, where his math implodes in the face of an simple physical experiment.

The final sentence of this book, should be the first:

Perhaps some day I'll be able to persuade a mechanical engineer to sit down and explain to a mathematician just how this problem really should be handled.

The problems all start in Lecture 1, which is so uncoordinated that it is incoherent. He is trying to deal with Newton's Laws. But they come out in ways that he thinks are amusing, like an object in motion tends to stay in motion in a straight line, which just also happens to be a circle of infinite radius. I do not see how this concept can be of any use.

I am going to take a wild guess at why Spivak's writing and methodologies are so disorganized. He makes constant references to stores and locations in Tokyo. Perhaps he lives there and has taken an academic position. If so, he has surely been exposed to Japanese technical writing, perhaps even gone native. These Japanese technical writing structures can seem terribly convoluted to Western scientists but seem perfectly natural to Japanese scientists. But i will not speculate further since i cannot confirm his residence in Japan.
posted by charlie don't surf at 12:05 AM on October 5, 2014 [1 favorite]

JackFlash - I don't buy what you're saying at all about being able to see the difference at the top and bottom of the Tokyo Tower. The difference in gravitational acceleration at the top and bottom of the tower is roughly 0.001m/s^2. With a 1kg mass, and a spring constant of 1, we are talking about a difference of about 1mm; as far as I know, that's pretty low for a spring constant too, which is going to only make it harder to see.

On the scale of a building, and certainly on the scale of your kitchen table, you will not see a difference, unless you have incredibly sensitive measuring equipment.
posted by vernondalhart at 1:15 AM on October 5, 2014 [1 favorite]

The "censored" bits of Newton that we're going to "come back to later" make this a bit tiresome to read. Also, it also seems like the author is going out of his way to "not understand" obviously intuitive physics concepts like density and rigidity. You may not have a fully rigorous mathematical description of the rigid body on day 1, but that doesn't mean you don't "understand" what rigidity is. Physics (unless you are doing string theory!) proceeds differently than mathematics: the absolutely rigid mathematical definitions (if they are ever found!) are the end-product, not the starting point, of physics. It starts from our physical experience of the world (aided by careful experimentation), not from a set of axioms. A "Poincare-like pragmatism" (to borrow a lovely phrase from another commenter) is not only helpful for differential geometry, it's essential for physics! (And, an Einstein-like imagination doesn't hurt either).
posted by crazy_yeti at 5:19 AM on October 5, 2014

Why are math books like this so expensive? Vast numbers of good books on 'the zon' are basically the cost of shipping but jeepers, $64 for a questionable used copy! And that seems cheap as math books go.


Second, is JackFlash's comment...he lost me right near the beginning when he talks about "uniform gravity"." the same as the argument that there can never actually be a triangle? As in, no diagram or picture of a triangle is actually a triangleIt, even if drawn down to an atomic level? The ends would always be in some way rounded and never an actual 'point'?
posted by sammyo at 5:50 AM on October 5, 2014

Also, it also seems like the author is going out of his way to "not understand" obviously intuitive physics concepts like density and rigidity. You may not have a fully rigorous mathematical description of the rigid body on day 1, but that doesn't mean you don't "understand" what rigidity is. Physics (unless you are doing string theory!) proceeds differently than mathematics: the absolutely rigid mathematical definitions (if they are ever found!) are the end-product, not the starting point, of physics. It starts from our physical experience of the world (aided by careful experimentation), not from a set of axioms.

I think that's the point of the "for [some] mathematicians". In mathematics, there's a large space between thinking you have an intuitive understanding of something and saying things that are true. It doesn't matter if I have an intuitive notion of 'density'. I want to know that your notion of 'density' and my notion of 'density' are the same thing.
posted by hoyland at 8:15 AM on October 5, 2014 [1 favorite]

Also, it also seems like the author is going out of his way to "not understand" obviously intuitive physics concepts like density and rigidity. You may not have a fully rigorous mathematical description of the rigid body on day 1, but that doesn't mean you don't "understand" what rigidity is.

These "obviously intuitive" concepts have always been the hardest part of doing physics for me. Can you give a short description of how a rigid body works in classical mechanics? A piece of string? More often than not, stuff like that is the key to understanding a problem, but nobody bothers to spell it out: it's mostly handwaved away as "intuitive", or students are just taught that this is the way to solve this problem because that's the way you solve this problem and do not even understand there's a bunch of underlying concepts.
posted by Dr Dracator at 8:15 AM on October 5, 2014 [5 favorites]

He makes constant references to stores and locations in Tokyo. Perhaps he lives there and has taken an academic position.

I'm pretty sure Spivak dropped out of academia decades ago and doesn't have an academic position. He says in the beginning (as in the first page or so) that he's writing up from lecture notes from some lectures he gave in Tokyo while visiting there (which probably means a visit of several weeks or months).
posted by hoyland at 8:24 AM on October 5, 2014

Gravity is uniform in the same way that speeds are 'slow' and space is 'flat'. You make all kinds of assumptions on Newtonian mechanics to make the math work out nice.
posted by empath at 10:49 AM on October 5, 2014

Proving I at least skimmed the PDF:

"... in addition, the iron becomes special at the same time, something we
may not usually acknowledge. If we take one of those cute little magnets that
people use to hold notes on a refrigerator door, and hold it near the refrigerator,
we don't notice the refrigerator moving toward the magnet! Instead, we feel the
magnet being pulled toward the refrigerator. Nevertheless, people don't usually
go around saying that refrigerators attract magnets. What's truly amazing, of
course, is that not only does iron attract magnets, but it does so in exactly the
right amount so that conservation of momentum holds.

Remarkably enough, our modern understanding of magnetism happens to
make this wonderful reciprocity quite understandable: the individual atoms of
the iron each act as magnets, except that they are oriented randomly, and the
magnet causes them to align, so that the iron now acts as a magnet also. So,
ultimately, it's all a matter of iron atoms attracting each other, and we have a
completely symmetric situation.

That same argument, of course, would make it quite clear why two objects
that have been given static electric charges should exert forces of equal magni-
tude on each other, since ultimately it's all due to the mutual repulsive forces
between electrons.

Strangely enough, it's actually the most common example, the repulsive force
of colliding bodies, that now seems the most mysterious....
posted by hank at 3:46 PM on October 5, 2014