The notion of "integrability" is everywhere in physics these days. It's a hot topic in high energy theory, atomic physics, and condensed matter. I hear the word at least once a week, and every time, I ask the speaker what precisely they mean by it. But I've never gotten a satisfying answer.
In fact, nobody even seems to be willing to say anything that integrability is or is not, they only tell me that it's associated or not associated with other vaguely defined notions. I've been told:
- integrability is sometimes associated with having a closed form solution
- integrability is sometimes associated with being "nice"
- integrability is sometimes associated with having infinitely many conserved quantities
- integrability is sometimes kind of like the opposite of chaos
- integrability is sometimes kind of like the opposite of thermalization
In every case I have responded by asking "so is that the definition of integrability?" and received some noncommittal mumbling in response. That is, nobody I meet who talks about integrable systems can state the definition of integrability. For example, the Wikipedia page linked dances around giving an actual definition of an integrable system, and when it does actually define them, it provides multiple different definitions, which run the gamut from being so weak they're meaningless, or so vague that they aren't definitions at all, before quoting a physicist saying "if you gotta ask, you'll never know".
I know that there exists some notion of integrability in classical mechanics, but I'm not sure if it's general enough to be linked to all five meanings here -- not to mention that most discussion I've heard of integrability has been in quantum systems.
So, as directly as possible, what is the actual definition of integrability used here? How is it linked to all of these vague ideas?