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Re: [Phys-l] Classical Adiabatic Invariant




On 2011, Jun 08, , at 16:44, curtis osterhoudt wrote:


Despite our propensity for giving hints, I'll just post a relatively nice treatment of the problem:
http://www.scribd.com/doc/50526620/26/Adiabatic-invariant-of-a-pendulum

V.I. Arnold, in his "Mathematical Methods of Classical Mechanics" (Section 52 in the "2nd, corrected printing" I have access to right now), makes it very clear as to which parameters we may change and have Liouville's Theorem still apply. The fact that we can parametrically resonate the pendulum (your swingset example) is well-known, and requires knowledge of the phase of the pendulum. It turns out that ignorance of this phase is equivalent to a requirement that the parameter we change is changed in a twice-continuously-differentiable way.

It's a very interesting problem!


At least one of my google found references refers to Arnold -- perhaps it'll inform as to whether intermittently applied artificial g change satisfies the adiabatic requirement.

AND whether the pendulum must be "free" or may be "clock" driven. I've intermittently sinusoidally "g" modulated two clock driven pendula (Q difference ~ 10X) with 0.002=>0.02 Hz sweep (5,000 seconds) the respective amplitude changes are ~ 28 and 4 millirad; no modulation ~ 120 millirad.


bc must read above reference later; is off to demo day at Pierce College.