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Re: [Phys-l] Classical Adiabatic Invariant




On 2011, Jun 06, , at 12:14, John Denker wrote:

On 06/06/2011 10:53 AM, Bernard Cleyet wrote:
I'm attempting to understand this as applied to a simple pendulum.
The examples I've found apply to shortening the rod.

OK.

Wherein E/omega is a constant.

That may be true in selected examples, but it's not true in
general. See below.

Shortening the rod increases the frequency, so the E must increase.

I wouldn't have said "must" increase. Maybe the E increases,
maybe it doesn't. Much depends on details of how the
shortening is carried out.

Again increasing g increases the frequency and energy (work on the
field added), so the E increases again intuitive ("sort of").

That's not what my intuition says.

A proverb says "education is the process of cultivating your
intuition".

In this case, my intuition is based on experience with
_parametric amplifiers_ wherein some parameter such as length
or g is varied. Depending on the time history of the variation,
you can get any result you want, including changing g in one
direction while changing E in the opposite direction.

Sorry I was inexact! The changes are very slow. i.e. The increase of v. ~ 60 microgals, (in ~9.8E+8micro. ) which "takes" ~12 hours. The period of the pendulum is 2Hz.

The period and amplitude are measured w/ a sensitivity in about one ppm or better. The clock at the Naval Observatory (Shortt #41) shows this variation due to the moon and sun well above the noise. (This may be the the result of "averaging" over a period of days, I don't remember.)



http://www.leapsecond.com/hsn2006/pendulum-tides-ch2.pdf




Classical Adiabatic Invariant

The word "adiabatic" is used in the literature in a couple of
different ways, neither of which makes sense in this context.
The simple harmonic oscillator has only one state and has zero
entropy to begin with, so there is no point in talking about
corresponding states or talking about entropy transfer.



http://docs.google.com/viewer?a=v&q=cache:2D-1w-UPdF0J:www.camtp.uni-mb.si/socrates/marburg2004/robnik.pdf+theory+of+adiabatic+invariants&hl=en&gl=us&pid=bl&srcid=ADGEESh3bpMKh70ZUpO-wNPhnL0zMjqorIUC262_EwWDvDL93go5ehJ7GNIoI9bSkYGZ588Yix2yL6iogvJxyXdNS35kBhU0jKxDdEY_Id5Fzn4Dx4h2Xr3vdnsDPMNRj_xnTyTEyOhI&sig=AHIEtbR1EjeOBuJWOmT6D4R_-y4FLvqDqQ


bc thinks he's got it (not the theory, but some of the application), but not the amplitude.