Swing Away...

It's been a while since I've written code out of curiosity

In physics, you can do a lot with a pen and paper -- so much that in my seven years as a physics student, I've felled more than my share of trees. Many times, us physicists are trying to figure out how something moves -- so "solving a system" means writing down the equation that tells you where something is at different points in time.

So, you'd think that something as simple as a pendulum could be solved in a few lines of math... and you'd be wrong. Turns out, the equations that tell us where a pendulum is at any point in time are easy to write down, but you can't solve them without the help of a computer.

So, what did we do before we had computers? Same thing we do now -- make approximations to make progress. If I don't pull the pendulum back very far -- the small angle approximation -- the system can be solved on paper. In my undergraduate days, I was happy to accept the approximation and move on. But last night, I got curious and wrote a computer program that gives me the real answer (or, rather, as close as a computer can get to the real answer.)

The results are pretty interesting! In my simulation, the pendulum is pulled back 45 degrees and released from rest -- too far for the small angle approximation to "work." In the graph, you can see the non-linear solution that has to be done with a computer, and the approximate solution which is done on paper. The two are pretty close for the first couple of seconds, but the gap widens as time passes.

So, is our approximation garbage? When it's not appropriate -- yes. But let's see what happens when I only pull the pendulum back about 11 degrees:

That's better. When the angle is small, the small angle approximation works really well!

In high school and undergraduate physics, you spend most of your time coming up with exact solutions to approximate systems -- like we do with the small-angle pendulum. However, in graduate physics and research you have to make a massive shift in thinking. You look for approximate solutions to the real system -- like I did when I wrote my computer program. It takes a while to get used to this shift, and I think we can do more in high school and college to prepare students for it. A lot of that preparation involves learning how to use computers to solve problems -- something I wish I'd done far more of in years past.

So, if you are a high school or undergraduate student and you think you might want to make physics into a career -- learn to code. Learn to solve problems with computers in as many ways, and in as many languages, as you can. It is the single most useful skill that I have (and I don't even do it very well.)

The Nontrivial Drinking Game

Let's play a drinking game: As you sit quietly in your physics classroom, studiously scribbling notes and doing your utmost to keep up with the professor, take a long draw from your hip flask every time you hear the word "trivial."

Every. Time.

In my experience, you'd be passed out drunk by the end of a typical lecture.

What am I on about? In physics (and mathematics, and chemistry, and who knows how many other scientific fields) we have developed our own dialect. It may not be apparent at first, until pay attention to how often you run across phrases like "to first order...", "without loss of generality...", or "in the limit..." These all mean a great deal scientists, but very little to everyone else. They are terms of art.

But there are other phrases that creep into our dialect... far more insidious phrases. "By simple manipulations...", "after some trivial algebra...", "it is obvious that...", are all commonly used to describe things that are not simple, trivial, or obvious to most people. I've even had professors describe proofs and examples in class as "simple-minded."

Typically, we use these phrases to describe long, tedious algebraic manipulations that we aren't inclined to reproduce on the board or the page. By describing them as "trivial," what we really mean is "time-consuming" -- there are only so many minutes in the lecture, so much space on the page, and we'd much rather focus on the set-up and results than the greasy, grimy machinery in the middle.

But here be dragons, because it's in the middle where many students have the most trouble. "The middle" is where minus signs go to die, it's where factors of 2 drown their sorrows with that missing sock from the dryer. The algebraic manipulations, approximations, limits, and assumptions that form "the middle" of many derivations in physics are far more complicated than we make them seem with words like "trivial", "obvious", or "simple-minded."

This may seem like semantics -- but there is real harm to be done. For a student struggling to understand a concept, hearing that it's "trivial" is hugely demoralizing. It's a taunt. Even now, after seven years of studying physics I struggle with the feeling that I shouldn't be here, that I'm not good enough to have advanced as far as I have: Things aren't as trivial as everybody else makes them out to be.

They call it "the impostor syndrome," and I'm certainly not alone in feeling this way.

What's more -- we lose nothing by eliminating words like this from our parlance. Saying a problem or bit of mathematics is "trivial" only serves to alienate those who struggle to understand it. It provides no useful information whatsoever. It is, in fact, quite the opposite of teaching.

I'm certainly not the first person to rail against these words and phrases. Matt Landreman wrote an excellent opinion piece in the pages of Physics Today titled "A Nontrivial Manifesto" that I encourage you to read. The only reason I am adding my voice to this matter is that we are far from resolving it. Seven years since that article was published, I still hear these words spoken from the blackboard.

We learned to speak this way, and it's high time we un-learned these phrases. We have to wipe them from our unique dialect, or continue to suffer the consequences. Physics is a non-trivial matter -- that's what makes it so damned interesting. There is tedium, but it's skilled, worthwhile tedium. We should acknowledge, not dismiss, the hard work involved in understanding our subject and developing the skills to master it.

If you've read this far, I hope you take my words to heart. If you are a graduate student, teacher, or professor - do your best to police yourself. Print off Landreman's excellent article and circulate it around your department. If you are a student, talk to your professors about how unhelpful words like "trivial" are in the classroom. If we work together, I believe that small changes like this will make the physics class room that much less intimidating