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Re: [Phys-l] Classical Adiabatic Invariant

This is not dimensional analysis, nor did I argue that it was (though my writing WAS imprecise, and though I wrote that the energy is proportional to something-or-other, I didn't mean that we don't know the prefactor). Nor did I argue (or support any arguments  -- just made them -- though these are well-known results) that I was replacing anything with max(p) or max(x) or RMS values thereof or anything like that. However, with the assumption that we vary the length of the pendulum smoothly (without reference to its current position or velocity, say), bc's original statement about E/omega is just fine.


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!





 
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________________________________
From: John Denker <jsd@av8n.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wednesday, June 8, 2011 2:46 PM
Subject: 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.
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