408:...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)
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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
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by Binney and
Tremaine, a constant of motion is a function of phase-space coordinate (position and velocity) and time, and an integral of motion is a function of only the phase-space coordinates. Every integral of motion is a constant of motion, but the converse is not true. I will insert this
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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". :-)
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reveals that they either explicitly says they are synonyms, or they make the subtle distinction of using "integral of motion" for the expression that is constant, and "constant of motion" for the value at which it is constant. In either case the two articles should be
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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
316:. 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?
472:; without checking Goldstein, I do not recall any constants of motion that cannot be expressed as an integration constant, so they might be exactly synonymous. I say wait a few days and then go for it. -
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Can't an integral of motion have an implicit time dependence whilst a constant of motion not? I don't think these two should be merged.
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article is too abstract; but perhaps you might have the confidence to write a simplified intro for it?)
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There seem to be several chaotic systems in which energy is not conserved, e.g., the driven pendulum?
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are synonyms. So they should be merged. Question remains, what should be the main article? (
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