## 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.

## Visualising Linear Algebra

I was recently in a physics education research meeting where somebody commented that it was impossible to visualise abstract linear algebra, or words to that effect. I objected that I do visualise linear algebra, and being the only one of this opinion (probably also the only one who would argue that maths is made of the same stuff as poetry) I was asked to explain. I stumbled through a few unconvincing sentences before giving up, but I thought I’d try it again now that I’m not so much on the spot. Of course, putting things into words both adds to and takes away from the idea in my head, but perhaps this will be interesting, if not useful.

Floating in a void of blackness is a point of cyan light. The colour is not terribly important, but it does identify the vector – this point of light is a vector – as belonging to a particular vector space. A vector from another space might look nearly the same, but it would be red, or green, or some other colour. Now that I’ve identified the void as a vector space, I realise that of course it isn’t empty: there’s not just one, but many, infinitely many, points of light floating there. (This reminds me that finite vector spaces are probably a thing, although not of interest to most physicists, and irrelevant right now. There are infinitely many vectors in our vector space.)

It would be impossible to draw a picture of this: infinitely many points of light blurring together so that they fill all of space, but entirely distinct, each one contained within itself. Fortunately the visual cortex appears to be somewhat more flexible than the retina. Since this is a vector space, and not a less orderly algebra, the points of light are neatly lined up in rows and columns – up, down, right, left, backward, forward, and more directions that are hard to describe. Most interesting vector spaces have more than three dimensions. I can usually only see three or almost four of them, but I have the uncomfortable sensation that one or two or infinity more are lurking behind me, watching. They’re smug, not malicious, but as an undergraduate student I had at least one nightmare about vector spaces.

Enough about the vector space as a whole – what happens if we look at an individual vector? Well, technically we can’t see any detail just yet. We need to choose a basis that pins the dimensions into place so they can’t lurk behind me or spin around each other just for the fun of it. But snapping a basis into place, we can zoom in on one of the distinct-but-continuous points of light. It looks a little like a planet covered in skyscrapers, or like the output graph from a spherical LHC calorimeter. There’s one skyscraper for every dimension; I can’t get all the skyscrapers into focus at the same time, because I’ve decided that this vector space is a Hilbert space, and has infinite dimensions. I can move along the vector space in any direction, and watch the skyscrapers change. Just one skyscraper changes with each hop, becoming a little taller or a little shorter.

While we’re looking at this vector, we can pull in another one, with an entirely different (or, for that matter, identical) skyscraper pattern. Smushing them together is the process of vector addition. Some skyscrapers reinforce and grow taller. Others cancel each other out, because skyscrapers can be either positive or negative – dark or light – and while I often ignore the distinction, it matters in this case. As the new vector shimmers into existence there’s a gentle pop and we teleport to a new location in the vector space.

I’ve taken a fancy to this vector, so I decide to change the basis to make it look special. As the new basis slides into place, the skyscrapers here all merge into one incredibly tall tower. It’s easy to pull in other vectors and compare them, now: how prominently does this megatower feature in their landscapes? Technically, I didn’t need to change the basis to calculate these inner products, but this is far more convenient than comparing each of the infinite skyscrapers on each vector.

When I drop the constraint of the basis, I’m forced back out into the sea of twinkling cyan that makes up the vector space at large. But the relationships I found before still hold, although it’s hard to see that here. I can start to find patterns, like golden threads, that flex with the vector space as the dimensions shift and change, but remain as markers of a complicated kind of order I couldn’t see before.