## Mechanics and the quest for elegance

Classical mechanics (the study of forces and motion at ‘everyday’ speeds and sizes) has, in the past, seemed like a rather clunky subject to me. It’s the branch of physics concerned with answering such questions as, ‘If I fall down the stairs and roll all the way down, how fast will I be going when I hit the floor and how hard will I hit? What if I slide instead of roll?’ and ‘If a rocket explodes into several pieces mid-flight, where can you expect the pieces to land relative to the rocket’s original destination?’ Not exactly the epitome of elegance (although reframing the unfortunate stair-climber as a performing artist and the ill-fated rocket as a firework might help a little).

So I was a little surprised at myself when I heard about someone’s dislike for classical mechanics and thought ‘But it’s so elegant!’ When did I start thinking that bashing rocks into each other was elegant? Apparently time and teaching* physics will do that to you, since I’m pretty sure I didn’t graduate with that attitude. I knew mechanics was useful, but I wouldn’t have described mechanics as a quest for elegance. Recently, however, I’ve needed to think about introducing not-so-introductory students to the idea of mechanics beyond Newton’s laws. And my thinking about the different ways of formulating mechanics is settling down around this idea of a quest for elegance.

Perhaps I should back off for a moment and explain what I mean by elegance in physics. Mostly, it’s about the idea of being able to express a complicated idea in a simple way. For instance, if I say ‘I love my dad,’ (Hi Daddy!) pretty much everyone immediately understands what I’m saying, even though I’m describing a complex relationship with lots of details and intricacies, which the statement glosses over. To give a more physics-y example, the standard model of particle physics lets us describe every known substance in terms of three basic families of particles (and just seventeen types of particle in all). This elegance makes it incredibly useful for explaining how our world works.

Newton’s laws were revolutionary in his context, because they described all kinds of motion in just three simple rules. Well, sort of simple rules. Actually, despite what we learn in high school, Newton’s laws are an awful mess. For instance, his second law is supposed to tell us how forces relate to motion. But when we get to Newton’s second law, we haven’t yet said what a force is. So we have to decide whether this law tells us what a force is or what a force does. It doesn’t contain enough information to do both. For practical purposes, we often have a good a good enough idea of a force as a push or a pull to make things work without exactly defining force. (This would probably be called the ‘shut up and calculate’ approach if we were talking about quantum theory.) But most philosophers of science seem to think it’s more consistent to take Newton’s second law as a definition of what a force is, which may be more philosophically satisfying, but isn’t terribly helpful for producing answers.

I don’t think anyone’s ever come up with a really good answer for what a force is. I know some physicists who say we’d be better off scrapping the idea of forces altogether (although I suspect this may be more for the sake of debate than anything else). But people have come up with cleaner rules that Newton’s. Some of them require fairly advanced maths (like multidimensional calculus), so we don’t tend to see them in school-level courses. Others are more familiar, at least in their basic forms.

For instance, it’s widely known that energy is conserved. A high school physics student can probably even use that fact to solve (parts of) the falling-down-the-stairs problem I posed earlier. Less well known is a theorem proved by Emmy Noether, which shows that energy conservation is intimately related to the fact that the laws of physics don’t change over time. (Thank goodness they don’t.) And the fact that momentum is conserved is intimately related to the fact that the laws of physics don’t depend on position. Of course, the results of the laws will depend on position — your weight is different on the moon and the earth — but the laws themselves don’t change. You do the same kind of calculation to work out your weight in each case. Weirdly, the laws of physics do change if you create a mirror-image reflection of a situation, but that’s a story for another day. (Look up the Wu experiment for more on that.)

Noether’s theorem very neatly ties together topics that at first glance seem completely unrelated. That’s elegant! In fact, Noether’s theorem is so well-made that when we start talking about topics like quantum theory and have to abandon Newton’s laws, Noether’s theorem and the conservation laws remain perfectly good to use. The same is true of other classical mechanics tools, like Lagrangian and Hamiltonian mechanics, although they do need a little fiddling and retuning.

I’m not sure I’ll ever be fascinated by the classic rock-hits-rock story of classical mechanics, useful as it is. But behind that story is a story of trying to describe and explain some of the most basic things in our world as well as we possibly can. And engaging with that quest for elegance is a genuine delight.

*People sometimes assume that since I teach, I must be in charge of directing a whole class in learning a subject.  I get to do that occasionally, but far more often my teaching looks like writing up model answers to a homework assignment or asking pointed questions about how a lab group has set up their experiment. Just so you know.