Stuff

My blogging has been a bit sparse lately. It's ironic, though, because I've been thinking about a lot of blog-worthy topics lately. I think that my problem is that I've been arriving at more questions than conclusions. So this is one question.

I'm taking a course on classical mechanics this quarter. The main focus is on Lagrangian and Hamiltonian formalisms in physics, as well as some related topics like the calculus of variations, chaos, and rigid-body motion. The relavant point is that the alternative formulations of mechanics are entirely equivalent to Newton's laws. In fact, Newton's laws can be derived from the one simple postulate and series of transformations of Lagrangian mechanics. If these formulations are indeed equivalent, it raises the interesting question: how do we decide the order in which to teach them? The current state of affairs is to preach the Newtonian formulation as a mantra until the second year of the undergraduate physics curriculum, at the very earliest (less demanding physics programs may even postpone this until the third or fourth year).

Answering this requires that we define differences between the formulations and evaluate their pedagogical effectiveness. I will restrict myself to Newtonian and Lagrangian formulations, because that is what I know best. Newtonian mechanics is centered around the notion of forces causing acceleration on bodies (F=ma), from which the equations of motion may be calculated. I find two important characteristics in this view. Firstly, the form of the equations conceptually suggests that motion is determined by some sort of external agent on the system; the second law implies a causal mechanism. Secondly, Newtonian mechanics is coordinate-dependent. Consequently, it is invalid in noninertial reference frames, and has an explicit reference to the coordinates imposed on the system. The primary pedagogical implication of these two observations is that Newton's laws lend themselves to being intuitively obvious but rather tedious and awkward to use for all but the simplest situations. Even someone with an undeveloped physical intuition can understand the meaning and application of the Newtonian formulation. But this same simplicity, I think, can ultimately be limiting. The notion of what a force actually "is," I think is somewhat lacking. True, force is something proportional to acceleration, and I apply a force when I push you, but that doesn't really say much. What makes force important? It seems aesthetically undesirable to define our mechanics in terms of some quantity that we simply define for the job. The result is that students think of mechanics in terms of these superficial causal mechanisms that just happen to work for some reason, rather than being a result of the structure of space.

Lagrangian mechanics, on the other hand, is much more abstract and useful than Newton's laws. It has one postulate: the path that any particle takes is such that the action is minimized (where action is an integral of kinetic minus potential energies over a path). At first, this may seem more arbitrary than Newton's simple laws, but it seems that this postulate ultimately says something about the nature of space, whereas F=ma does not. Considering the transformations of a particular space, conservation laws follow immediately, and are not introduced in a sort of deus ex machina fashion. Furthermore, there is the advantage that it is, in some sense, independent of coordinates and reference frames: it works in noninertial systems and arbitrary coordinates, without much trouble. It also leads to tremendous simplification of calculations. From a pedagogical point of view, the downside is its analytical complexity: although fairly straightforward, it requires more advanced mathematics than the laws of Newton.

It seems that the teaching of physics follows a primarily historical development. We first teach Newton's laws, then E&M, then Lagrangian mechanics, then more advanced E&M, then quantum mechanics, then some miscellaneous modern topics, then after a review of basic physics in the first year of grad school, quantum field theory, string theory, and the latest developments. On some level, this makes sense. If man discovered physics roughly in this order, then the student will be able to learn everything in a perfect and logical progression from the historical approach, if you throw out a few of the logical inconsistencies. Everything will follow naturally from everything else. But the real question--the biggest question of all: is the historical approach the best? Strictly speaking, I see no reason why this answer should be yes. If our goal is to train physicists, shouldn't we be trying to force them to think the way physicists actually think? Doing this requires that physical intuition about symmetries, fields, and other important things be stressed from the very beginning, rather than slavish devotion to calculation of uninteresting and unmeaningful problems. There is this problem in physics education, where each class tends to teach something that contradicts the previous class. Special relativity contradicts classical mechanics, quantum mechanics contradicts both, QED supercedes quantum mechanics and electrodyanmics, and so forth. The historical way theories were developed requires that this always be the case to some extent. But it shows how the physical intuition developed at each level leaves one unprepared for the next level.

Considering these observations, it seems theoretically more desirable and useful to teach Lagrangian mechanics first, or at least treat Newtonian mechanics as secondary to it. But there is the practical problem of mathematical complexity. It is unreasonable to expect any more than a handful of high school students to comprehend the calculus of variations. Since most people take their first physics courses in high school, our program appears doomed to failure. But most people only get serious about physics in college, so we will limit ourselves to that stage. I see no reason why there could not be a small class for highly motivated students of physics that starts immediately with Lagrangian mechanics. I am effectively in that situation right now. I was able to place out of the first-year honors physics course, and I started with a course on elementary quantum mechanics (which incidentally was extremely easy) and this current classical mechanics course. There are many first-year students at good colleges and universities possessing the mathematical machinery needed for such concepts. The rest of the students in physics could continue with their current Newtonian-based, historical program, which seems to currently work okay, and which might be more approachable.

Lastly, I will anticipate one major criticism of my plan: abstractness. All you boring people say: this is too abstract! As we teach children arithmetic, then fake algebra, then calculus, then analysis, then abstract algebra, etc., so it is with physics: we must start with the most concrete, then work our way up. You say: analysis is harder than doing those stupid integrals from calculus, so it must be taught later. But is it really harder? Or is it just harder because your mind has been indoctrinated from day 1 to think of all math in terms of 1st grade arithmetic: if I follow a set of rules, I will calculate the correct result--the end? This is the general question I have been thinking about lately: why do educational programs move from concrete to abstract? Can people learn something by starting with abstract concepts, and developing the concrete applications later on? For a lot of aesthetic and practical reasons, I think they should if they can, but I don't know if it's possible. I really have no idea what the answer to this question is, nor how to answer it.

To finish off this discussion, here is one piece of anecdotal evidence that I have thought about a lot. It turns out that the University of Chicago's mathematics department is a fantastic place to be an undergraduate, for many reasons. One of these reasons, I think, relates to abstractness. Each year, all first-years take this mathematics placement exam that covers everything from arithmetic to analysis. The students who are very good at math and have at least some experience with basic calculus are placed into this year-long course colloquially called "the 160s" (i.e. honors calculus). From the very start, they only study the theoretical aspects of one-variable analysis: sorts of things like sequences, compactness, continuity, integration, Lebesgue measure, etc. Everything beyond the 160s is similarly abstract. In other words, abstractness is taught from day 1. People do quite well in such an environment; they tend to love it. It is a revealing fact that the U of C has more undergraduates who major in math than in english. This is opposed to most places, which have this battery of computation-centric courses on differential equations, multivariate calculus, and linear algebra, which are sometimes fun but not very abstract. They're useful for things like physics and engineering, but its the kind of thing that any competent physicist or engineer should be able to pick up "on the street," so to speak.

I'm not sure if that says anything definitive about my final question, but it suggests that abstractness isn't always unnatural or counterintuitive to the unintiated mind.