My Thoughts Precisely

I've been thinking a lot recently about what's happening to my interest in physics. I've now begun my second year of four and half post-Deep Springs years towards a Masters in Engineering Physics. This fall I'm taking Systems and Transforms, Mechanics of Materials, Algorithms and Data Structures, Statistics, and Thermodynamics. Thermodynamics hasn't begun yet and Statistics just began this week, but I just wanted to provide some background for the discussion of mathematical physics/mechanics that follows.

The thing that lights my fire when it comes to physics is the idea of deterministic, definite understanding of the universe. Yes, quantum mechanics makes the foundation of many processes decidedly indefinite, but the foundation is still has a definite(ly indefinite) form. And when the processes are observed at a larger scale, brownian motions average (by definition!) each other out and you can apply models that work very very well for 10^23 particles. That is absolutely beautiful. Definitely beautiful. Snell's Law from optics is a simple yet elegant example of this.

But the statistical average behavior that much of classical physics models is far from the modeling used in engineering mechanics, which I protest against as categorically different. What got me thinking enough to write this piece is the equation for the linear strain at a given angle, phi.

E_phi = E_x*cos(phi)^2 + E_y*sin(phi)^2 + Gamma_xy*sin(phi)*cos(phi)

Don't bother studying the equation, what I want to discuss (but won't present in full) is the "proof" of this relationship. What I wanted to get across is, in spite of imprecisions, it's not exactly the simplest relationship. Yet it holds well enough and is incorporated into all sorts of methods for calculating principal stresses, which in turn builds the bridges and buildings that we all marvel over. But how does one arrive at this gem?

Well the proof, like the proofs for many equations in geometric mechanics, starts by assuming Gamma << 1, which gives

cos(Gamma_xy)=1
sin(Gamma_xy)=Gamma_xy

In my opinion, this is the first mistake. This is a radically different step than the statistical averaging in quantum mechanics. Here you are assuming a linear relationship where the true relationship is trigonometric. The next steps are too verbose to type out here, but after expanding a few squares, products of small numbers are discounted as negligible, and a series-expansion is done to only the first term to give the final result. These steps, in my strict view, leave something to be desired for the same reason.

It may be good enough, it may be very very good enough, but it's missing the golden nugget that so fascinates me about physics, the idea that models in physics describe the universe around us in the most exact way we know. Sure, Newton was wrong, but as the shortcomings of the Newtonian model became known, Einstein developed his relativistic model, and we once again understood the universe.

I guess I've talked myself into a bind, seeing as the Newtonian model is still useful for 99% of physics calculations, and they sure as hell don't use relativity in material mechanics. But I guess what I'm trying to say is that I'm not as interested in what's useful as in what's true. I guess what I'm trying to say is that I'm a physicist, a mathematician, not an engineer.

Perhaps I would be happier with my material mechanics (core) course if I had taken a later numerical analysis course first and could throw all this into MatLab and easily calculate the actual difference and see how negligible it really is. But the problem isn't that I don't understand the steps taken, I do, but it's the fact that derivation of the equation is completely blown away and the equation is all that remains on the equation sheet. And that's the root of what bothers me. I prefer to make my assumptions on my own so that I can see them, not have them built into my equation sheet where I can't understand them. I want to be able to assume absolute precision on behalf of any equation I pull off of an equation sheet. Also, the sheer multitude of equations in material mechanics means that I can't keep track of these approximations, and that gives it all a general plug-and-chug feeling. (The practice of looking up material properties in endless tables doesn't help).

The other day in my Systems and Transforms (which contains complex analysis) class the lecturer was proving Cauchys Integral Theorem (apologies if all these translations come across wrong). And he said, OK, we are going to use Greens Equation at this point, and used it, and everyone understood how and why. Now, we spent several weeks working with Greens Equation in multivariable calculus last winter, I'm comfortable with it, I even vaguely recall deriving it on the final exam. I can't say anything of the sort about any of the equations in material mechanics.

All these thoughts feed into a major vein of my current thought-flow, the possibility of switching majors to Applied Math. But at the same time, I've got this one shitty test in material mechanics in three weeks, and then the rest of the core is filled with much more interesting courses that are pretty much all worth considering as concentrations: control engineering, continual systems, applied atomic and nuclear physics, electrical measurements, electronic circuits, electronic materials, modeling and simulation in field theory, numerical analysis. After that (spring of 2007, yikes) then I'll be free to study all the non-linear maths I want.