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Re: [Phys-l] Classical Adiabatic Invariant



For each of your oscillators, there was a well-defined position (the crossing-point in your figure-8) at which the support length was changed. Each of them is being parametrically pumped at a particular point in their period. I don't see how that disagrees at all with what has been said so far. 


Are we perhaps disagreeing on the precise definition of an "adiabatic invariant"?




________________________________
From: John Denker <jsd@av8n.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Thursday, June 9, 2011 1:20 PM
Subject: Re: [Phys-l] Classical Adiabatic Invariant

If y'all want to understand what the Liouville theorem really says
about phase space in general, and harmonic oscillators in particular,
please refer to this diagram:
  http://www.av8n.com/physics/img48/phase-space-sho.png

It shows four snapshots of phase space.  The axes in phase space are, as
always, the position q and the momentum p, where p and q are canonically
conjugate.

We start with the upper-left snapshot.  This shows an _ensemble_ of 8
harmonic oscillators.  They all have the same energy.  They have 8
different phases, evenly distributed, as shown.

The equation of motion says that each and every oscillator follows a
circular path in phase space.  So they play follow-the-leader around
the circle.  The upper-right diagram shows the situation if we take
a snapshot 1/36th of a cycle after the upper-left snapshot.  Everything
is rotated 10 degrees.

Things get more interesting if we modulate the length of the pendulum
in accordance with the scheme diagrammed here:
  http://www.av8n.com/physics/img48/dpa.gif
Specifically, as indicated by the gray figure-8 in the diagram, we
lengthen the string at the end of each half-cycle, and then we shorten
the string at the middle of each half cycle (doing work against centrifugal
force).  We do this gradually but persistently.  If we do it at just the
right phase, we pump energy into the system, as shown by the red and cyan
dots in the phase-space diagram, i.e. the dots that start out at the 3:00
and 9:00 positions.  OTOH if we pump at exactly the "wrong" phase we suck
energy out of the system, as shown by the dark green and dark blue dots,
i.e. the dots that start out at the 12:00 and 6:00 positions.  If we
perform such pumping on an _ensemble_ of oscillators, after a while we get
a new ensemble, as shown by the lower-left snapshot.

We now discontinue the parametric pumping.

Note that the area spanned by the _ensemble_ of oscillators is invariant
with respect to the parametric pumping operation.  The initially circular
distribution of oscillators has become an ellipse, as shown by the dotted
line in the lower-left snapshot.  The new distribution is 2x smaller in the
p direction and 2x larger in the q direction ... so the area is the same. 

  This is an example of a Bogoliubov transformation.  That is, essentially
  we have just redefined the units on the q axis and redefined the units
  on the p axis to match, so that the new p and q are canonically conjugate.
  If this doesn't mean anything to you, don't worry about it.

As always, the equation of motion tells us that each oscillator rotates
around the origin in phase space.  The lower-right snapshot shows a
snapshot 1/36th of a cycle later than the lower-left snapshot.

It would be very, very wrong to think that E/ω is invariant.  For some
members of the ensemble, E increased very substantially.  For other
members, E decreased very substantially.  All that is without any
significant change in ω.  At all times the parametric modulation was
small and infinitely differentiable.

Let's be clear:  Note the contrast:
-- Area in phase space is conserved, if we are talking about the area
  marked out by the _ensemble_ of oscillators.  This is exactly what we
  would expect in accordance with the Liouville theorem.
-- For any particular oscillator, E/ω is not invariant.  Not even close.

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