Gas station without pumps

2011 June 9

Classical mechanics, with computer simulation

Filed under: Uncategorized — gasstationwithoutpumps @ 13:23
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In the comments on my post on Project Euler, plam pointed me to a physics book: Structure and Interpretation of Classical Mechanicsby Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer (MIT Press, ©2001).  The preface of the book explains the premise nicely:

Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dynamical systems. We concentrate on developing general methods for studying the behavior of systems, whether or not they have a symbolic solution. Typical systems exhibit behavior that is qualitatively different from the solvable systems and surprisingly complicated. We focus on the phenomena of motion, and we make extensive use of computer simulation to explore this motion.

The book appears to be all on line, so I may try to learn some physics from it.

The computation is expressed in Scheme (a LISP dialect), which is an understandable choice given that the book was written at MIT at a time when the first computer science course there was taught using Scheme.  The authors make a big deal about how little syntax Scheme has, which I do not necessarily see as a good thing.  Having to count positional arguments and parentheses all the time does not lead to greater readability and debuggability of programs.

Their point

The advantage of Scheme over other languages for the exposition of classical mechanics is that the manipulation of procedures that implement mathematical functions is easier and more natural in Scheme than in other computer languages. Indeed, many theorems of mechanics are directly representable as Scheme programs.

is well-taken, though, so I will withhold judgement on whether Scheme is a suitable choice pedagogically until I’ve had a chance to read some of the book.

Judging from the first few pages of the book, this text is not intended for students to learn physics from for the first time.  There seems to be an assumption that the reader is already familiar with the differential equations that are used in Newtonian mechanics, so that the different “variational” approach used in this text can be contrasted with the Newtonian approach without ever explaining that approach.

The first equation in the first chapter is already fairly opaque, with $\gamma$ as a path, $\mathcal{F}[\gamma]$ as a function of time that measures some local property of the path $\gamma$, and the unexplained $\mathcal{S}$ presumably being defined by this integral.

The first equation of the book is already pretty opaque.

I think I can work my way through the notation, but I doubt that I would give this book to a high school student for self-study.  I just hope that the mathematical elegance of the Lagrangian approach justifies slogging through some rather turgid prose.

1. You bet the Lagrangian formalism will be worth it. It’s really amazing that all you have to do is write down the kinetic and potential energies of something and then be able to model its dynamics. Fun stuff.

Comment by Andy "SuperFly" Rundquist — 2011 June 9 @ 13:42

2. The intended audience is more along the lines of MIT graduate students (to whom the authors gave the corresponding course) than high school students, yes. I guess that some high school educators are trying to figure out how to apply that concept to more basic-level physics though.

Comment by plam — 2011 June 9 @ 13:43

• Ah, that explains the rather large assumptions (that everyone already knows all the physics). Since I have not had physics since a non-calculus high school physics class in 1969–70, this book may take some doing for me to get through. I’m not too worried about the math or the programming aspects (though it has been almost as long since I did any programming in LISP), but the turgid prose with all the explanations buried in the footnotes may make it difficult for me to handle this book.

Comment by gasstationwithoutpumps — 2011 June 9 @ 13:51

3. If it’s helpful, check out my chapter 6 (calculus of variations) and chapter 7 (application to physics) screencasts here: http://www.screencast.com/t/Izof8VqlOn2 That’s intended for junior physics majors but maybe easier to digest than the book. Of course all of it is done with an eye towards Mathematica but maybe it could be of use.

Comment by Andy "SuperFly" Rundquist — 2011 June 9 @ 14:00

• Thanks, but the book will have to be really awful before I find screencasts or videos more appealing. I’ve never had much patience with the slow drone of video lectures. Books are much faster, and much easier to back up in if I realize I’ve missed something. I realize that a lot of students prefer videos, but I definitely do better with books—even ones with rather turgid prose and excessive footnoting.

Comment by gasstationwithoutpumps — 2011 June 9 @ 14:08

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