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Re: [Phys-l] Classical Adiabatic Invariant



On 06/08/2011 11:30 AM, curtis osterhoudt wrote:
The average energy of such a pendulum ~ [omega*position]2, and the
ratio E/omega

*is* an invariant (I'm not sure as to what you mean by "classical"),
based on the constancy of the phase-space-trajectory area traced out
by {x(t), (v(t)}.

1) We agree that phase space is conserved ... if we are talking
about the phase space of the whole system. Liouville's theorem
and all that.

2) We agree that you can argue on dimensional grounds that the
conserved quantity for the unperturbed simple harmonic oscillator
must be something like E / (m ω).

3) However, there is more to physics than dimensional analysis.
The actual statement of Liouville's theorem is a bit tricky.
It involves Δp and Δx as mapped out by neighboring trajectories.
You cannot safely replace Δp by the max p or RMS p, and you
cannot safely replace Δx by the max x or RMS x.

4) I insist that if you perturb the system, all bets are off.
In a complicated system, if you project out some little
one-dimensional piece of the phase space, that piece will
not in general be conserved.

This is important. Otherwise it would be impossible to build
things like refrigerators and heat engines and quantum non-
demolition measuring devices and zillions of other useful
things.

In particular, let's do the experiment: Go to the playground.
Select the longest (tallest) swingset you can find. Stand on
the seat of the swingset. Modulate the effective length of the
pendulum by bending you knees at the end of each half cycle,
and straightening your knees at the middle of each half cycle
(so as to do work against centrifugal force). The variation
in length is indicated by the light-gray figure 8 in the figure:
http://www.av8n.com/physics/img48/dpa.gif

This leaves ω substantially unchanged. I guarantee that you
can make very large changes in E and therefore in E/ω.

Indeed, this works so well that it is quite possible for E
to grow quite large, sooner than you might have expected,
so please be careful.

This is moderately interesting as a Gedankenexperiment, but
it is even more interesting as a real experiment. I beseech
to actually do the experiment. It's well worth it.

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.