# Practicing Scales and Arpeggios, Physics Style

Physics professors have this annoying habit. They’ll present a problem to be solved, figure out how to describe it in mathematical terms, and declare that the physics is done and the rest is just math.

Well yeah, but that’s kind of like drawing a blueprint for a house and then declaring to your trainee workers that the construction itself is a trivial exercise. At risk of sounding like Barbie, math is hard. Sometimes professors are particular drill instructors and they’ll do the physics part of the problem for you, leaving you to only practice the math.

But math is the language of physics, and to be able to do physics with any degree of proficiency you have to be able to do the math once you have the physical situation characterized properly. The goal of the professors is to make sure you get enough practice doing the math. I hope that’s their goal anyway. You never know.

In fact if I have any advice to give the undergrad physics majors it’s this: get good at the math. Really don’t skimp on the practice, or the math classes you choose to take, or the tricks and techniques you learn. Yeah Mathematica can do a lot of it for you these days, but really it’s less than you’d expect. It can do (say) a Fourier series for you, but you have to understand that you need a Fourier series in the first place, what it is, how it works, and how to actually carry it out. Mathematica can’t do much more than just make sure you get the actual integrals right.

For example, we have to solve this equation as part of a problem in the homework of one of my current classes. It describes the velocity of a projectile in a dilute medium subject to linear and quadratic drag. We’re supposed to find the velocity as a function of time.

That contains all the physics of the situation. The left side of the equation is just mass times acceleration, and the right side is the force. It’s just Newton’s laws. But you (or maybe just me) might feel an instant of creeping horror at the sight of that v2 term. Nonlinear differential equations are usually somewhere between tortuously difficult and absolutely impossible. But fortunately this one doesn’t contain time explicitly, and so it’s a special case called an autonomous differential equation. Despite its nonlinearity, those kinds of equations are solvable via the straightforward method described in the link. Given and initial velocity v0, the answer I get is here. I’m leaving it at the link in case any of you want to practice your ODE skills without seeing the result first.

And that’s an example of the kind of problem I was talking about above: there’s zero physics involved. The physics is figuring out what the differential equation is, but that was actually given to us in the problem. Drudge work? Yes. A pain in the neck? Yes. Very important to be able to do quickly and accurately? Double yes.

August 27, 2008

In my undergraduate classical mechanics class, the professor went one further. After arriving at a mathematical expression for the solution in terms of an infinite series, he told us that this wasn’t a good answer, because the terms were alternating in sign and comparable in magnitude, so trying to solve it on a computer would run into the problem that subtracting similar terms from one another greatly magnifies the error as a large number of meaningful bits cancel one another out. If you start with 1.0010+-0.0001 and subtract 0.9990+-0.0001, you go from a relative error of 1e-4 to a relative error of 1e-1. So, he declared the problem not solved yet, and went on to re-formulate the mathematical expressions in such a way that separate accumulators of positive and negative terms could first be summed, then subtracted without large loss of precision.

As far as I can remember, he was the only professor to talk to us undergrads about the practical application of the math to real-world computing hardware.

2. #2 Marc
August 27, 2008

*raises hand*

3. #3 Matt Springer
August 27, 2008

Don’t worry Marc, I’ve actually got some musical physics posts in the pipeline. Sorry to disapoint for the moment, but I promise they’re on their way!

4. #4 Frederick Ross
August 27, 2008

*raises hand* AND you threw that equation there with “solve” somewhere near it, and I was calculating the partial fractions for the integral before I knew it. Tsk tsk! Warn us!

But a solution in terms of well characterized families of analytic functions is often not the most revealing thing to do to such an equation. Get at the phase portrait and get some idea of the qualitative dynamics. It often turns out that many of the questions you might want to ask are encoded in the topology of the vector field just as directly as in the solution.

5. #5 Notkieran
August 28, 2008

That actually happened to me in my final thesis (a thermo-and-fluid-dynamic treatment of a supernova expansion instead of a taylor-sedov). Since I had been chasing false leads for quite a while and only had two weeks left to finish the thesis, I barricaded myself into my room and began to program an iteration sequence.

It was an Euler iteration.

In Excel.

I believe I referred to it later as a particularly physicist solution: in the absence of any more intelligent way to solve the problem in the time given, get a bigger hammer.