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.