Talk:Constant of motion

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[edit] Two Questions

Hi Linas, I was puzzled about two things you wrote and was hoping you could explain them.

(1) The first was that (system in which only energy is conserved) = (chaotic system).

• There seem to be several chaotic systems in which energy is not conserved, e.g., the driven pendulum?

Yes, you are correct, I was being sloppy.

• Chaotic systems need to satisfy properties that don't seem to follow necessarily from "only energy is conserved", e.g., orbits of initially close conditions diverge exponentially and become topologically mixed. Maybe it's too early in the morning -- am I missing something?

Well -- I have only a hand-waving answer. If there are not constants of motion, then the system is free to visit all phase space... and it will. If it does not visit all of phase space, then there is some constant of motion that you failed to recognize. If the system visits all of phase space, the question becomes only whether it is "merely" ergodic, or possibly more strongly mixing than that. I don't know any truly mathematical statements for this, but a pseudo-mathematical argument is that there are rigid theorems for expansive systems ("chaos is rigid"), which say "if it can occupy all phase space, it will, and furthermore, it will be hyperbolic in doing so." There are some more precise statements, but I don't know them by name. Axiom A pops into mind.

(2) The second was that non-integrable systems cannot be quantized consistently. I'm unfamiliar with the quantum chaos literature, but I would be surprised if there were any obstacle to applying the Feynman path integral method to a mechanical system with a finite number of degrees of freedom. Could you point me to a reference? WillowW 10:38, 12 June 2006 (UTC)

Just about any text in quantum chaos will make this clear. At a certain conceptual level, you may be right. At a practical level, tools, techniques and theorems are lacking. I've got one highly abstract paper that takes 100 pages to quantize the simple harmonic oscillator as a "simple example problem". :-) linas 14:52, 12 June 2006 (UTC)

Try Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (1990) Springer-Verlag. Very sharp and insightful, completely free of mathematical mumbo-jumbo, throughly approachable by anyone who's gone through undergrad physics. linas 14:58, 12 June 2006 (UTC)

[edit] Action-angle variables

I was also looking at your action-angle variables, an article I've toyed with writing myself for a long time. Good job. You may want to tie this article more strongly to that one: The constants of motion are the actions, and v.v.; both notions are at the core of the idea of an integrable system. (and the integrable system article is too abstract; but perhaps you might have the confidence to write a simplified intro for it?) linas 15:09, 12 June 2006 (UTC)

...And I looked at the other edits a bit; I see you shuffled some stuff off to the bottom. The point is not to get tangled by quantum chaos, which is an interesting but irrelevant distraction, but to instead make the statement "system with constants of motion == integrable system == system with symmetries" and conversely, "non-integrable system == system with no constants of motion". I don't know if there's a grand theorem that makes this statement, or what the name of this theorem is, but this seems to be the over-arching "great truth" of differential equations at this point in history. Furthermore, this connection should be put up-front and center to the article, not buried in its bowels. Unfortunately, I am no expert in this; I'm just repeating what all the experts say. 00:25, 13 June 2006 (UTC)

I'm digging around now. The key pieces are Frobenius theorem and Liouville's theorem (Hamiltonian). I'll see if I can come up with a more direct, concrete statement. linas 03:30, 13 June 2006 (UTC)