Afraid I’m behind on resuming Why Stuff Can Orbit, mostly as a result of a power outage yesterday. It wasn’t a major one, but it did reshuffle all the week’s chores to yesterday when we could be places that had power, and kept me from doing as much typing as I wanted. I’m going to be riding this excuse for weeks.
So instead, here, let me pass this on to you.
It links to a post about the Legendre Transform, which is one of those cool advanced tools you get a couple years into a mathematics or physics major. It is, like many of these cool advanced tools, about solving differential equations. Differential equations turn up anytime the current state of something affects how it’s going to change, which is to say, anytime you’re looking at something not boring. It’s one of mathematics’s uses of “duals”, letting you swap between the function you’re interested in and what you know about how the function you’re interested in changes.
On the linked page, Jonathan Manton tries to present reasons behind the Legendre transform, in ways he likes better. It might not explain the idea in a way you like, especially if you haven’t worked with it before. But I find reading multiple attempts to explain an idea helpful. Even if one perspective doesn’t help, having a cluster of ideas often does.