I’ve never been entirely comfortable with the concept of potential energy. Every time I’ve tried to understand it, the conversation goes something like this:

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Physics: Energy is conserved!

Me: Cool. What’s energy?

Physics: Well, there are lots of different kinds, but the main one is kinetic energy, which is \( {1\over 2}mv^2 \).

Me: Mass times velocity… squared? That seems like kind of a weird thing to be conserved…

Physics: It’s just something we noticed.

Me: But velocity squared? Does that even make sense?

Physics: Look, it’s been empirically verified. Anyway, so—

Me: And why only half?

Physics: What do you mean?

Me: What I mean is, if one half of mass times velocity squared is conserved, why not the whole thing?

Physics: Shut up, that’s just how we do things. Anyway—

Me: Hold on. So if I’ve got a one kilogram spherical cow, and let it fall for one second, it goes from zero energy to, let’s see, about 48 kilogram square meters per second per second—

Physics: Joules.

Me: Who’s Jules?

Phsics: Joules. The SI unit for energy. \( 1\, \mathrm{kg\, m^2/s^2} \) is a joule.

Me: Whatever, it still looks weird to me. My point is, that doesn’t look like \( {1 \over 2} mv^2 \) is being conserved at all.

Physics: Ah, so you see, while it gained kinetic energy it also lost potential energy!

(long pause)

Me: OK, I’ll bite. What’s potential energy?

Physics: It depends on the situation, but in this case, it’s mass times gravity times height.

Me: Mass times gravity times… OK, so if it starts off ten meters above the ground, it’s got 98 joules of potential energy plus zero joules kinetic, then one second after I let it go it’s got 50 joules potential and 48 joules kinetic?

Physics: Exactly!

Me: So what if I dig a hole?

Physics: Huh?

Me: If it starts off 10 meters above the ground, and I dig a 10 meter deep hole, then without doing anything at all to my spherical cow I’ve just doubled its potential energy.

Physics: Look, you can measure potential from the center of the earth if that makes you feel more comfortable. Just that the numbers get awfully big that way…

Me: I’m not buying it.

Physics: What do you mean you’re not buying it? It totally works!

Me: First you assert, without justification, that energy is conserved, but in the very first thought experiment I come up with it doesn’t appear to be at all. Then you tell me that everything also has “potential” energy that somehow balances the books, but that doesn’t even stay the same if I dig a hole. The whole thing sounds fishy.

Physics: Shut up, it works.

Me: Still not buying it.

Physics: You don’t have to take conservation of energy on faith, you know. It actually arises as a natural consequence of the Principle of Least Action!

Me: Oh, good—I was beginning to have my doubts. Show me how that works!

Physics: What you do is, you form the Lagrangian \( L(\dot{q}, q, t) \), integrate it over time to get the action, and then find the path that makes the action stationary: $$ \int_{t_0}^{t_1} {1\over 2}mv(t)^2 - mgh(t)\,\mathrm{d}t $$

Me: Whoah, where the heck did you get that from?

Physics: To be honest, I’m not entirely sure myself. Something to do with the calculus of variations, I think… But what’s cool is, you can analyze any system at all and use whatever coordinates you find convenient, just plug it into the Euler-Lagrange equations to figure out how it behaves!

Me: Just plug it into— ?!? Hang on, I don’t even understand how you got that first equation, it looks like you pulled it out of your ass!

Physics: There’s no set procedure for finding Lagrangians, actually. You just kind of get a feel for it after a while.

Me: In other words, you pull it out of your ass.

Physics: Hrmph. Well, don’t worry about it too much. Nowadays we mostly use Hamiltonian mechanics anyway.

Me: OK, give it to me. How does that work?

Physics: First you take the Legendre transformation of the Lagrangian—

Me: Oh, nevermind.

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It’s at this point that I usually give up and go back to studying something simple like category theory. But I recently read a book that finally gave me a clue. More coming up.