Re: [Phys-l] Classical Adiabatic Invariant

[Bernard Cleyet <bernardcleyet@redshift.com>]

On 2011, Jun 08, , at 13:46, John Denker wrote:

> I've done this experiment many many times over the years.
> You are going to have a very hard time convincing me that
> E/ω is invariant under the given conditions.
> __


The argument is: omega * tau is >> 1  Where tau "... is the time scale"    [s = t/tau]    for H(amiltonian) = 1/2 (p^2 + omega^2 (s) * x^2)  

Now --- I've "actually" done the experiment, but not yet analyzed, AND my method of artificially changing g may be invalid, because I don't have an Atwood's machine or parallel plate cap., etc.  I'm using a solenoid under a pendulum whose bob is (or includes) a very strong Nd magnet.  Above the solenoid is a coil, which is part of the pendulum's drive mechanism.  So the fields, of necessity to apply a force, are strongly divergent.  My calcs. indicate the  force on the Nd from the solenoid is essentially zero for most of the period.   So for that time scale it's about 1/2.  But my application of the force is tau = ~ 500    I find a sig. difference between pendula with quite different Qs (1), as intuitive.   My drive is a function gen.-amp.   0.001 Hz.   I've tried sine and sawtooth with the same general appearance of the position (angle) as a function of time graph.  The High Q differs by reducing the change in amplitude.   In regard to swing pumping, Baker and Blackburn (2) devote several pp. (42-44) to parametric forcing in reference to O Botafumeiro and revisits pp. 56-62 (swing and O. B. [includes second order correction - (angle^2)/16] 



(1) ~ equivalent simple pendula bob masses 0.076 and 0.911 kg
(2) "The Pendulum / a case study in physics"  Marian (5th ed.) uses similar techniques for non-harmonic drives.

bc's last experience with H was 55 years ago (Goldstein)

p.s. Goldstein's recent ed. supposedly quotes an exchange on photon number between Einstein and Lorentz of the pulled pendulum. (Solvay l)

and rigorous proof:

http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ9_3-4%2FS1446788700007254a.pdf&code=3ba2687143f8e8f5ae713db408d36660