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Re: [Phys-l] Classical Adiabatic Invariant



Once upon a time, there was a railroad car full of sand. The system
(car + the sand therein) was moving horizontall along the tracks with
some
initial mass m0,
initial momentum p0, and
initial energy E0.

The sand was very slowly leaking out, so that the mass of the system was
slowly decreasing over time.

A fifth grader was asked to use the principle of conservation of momentum
to calculate how the velocity of the care increased as the mass decreased.
He giggled and said "that's ridiculous". Everybody knows the car doesn't
speed up in such a situation.

Another fifth grader was asked to use the principle of conservation of
energy to calculate how the velocity of the car increased as the mass
decreased. She giggled and said "that's ridiculous". Everybody knows
that you can't calculate the energy (or momentum) without accounting for
the sand crossing the boundary of the system.

I mention this because at
http://tabitha.phas.ubc.ca/wiki/index.php/Phase_Space
we have a detailed calculation using canonical mechanics to "prove" that
E/ω is invariant for a pendulum where the mass is changing because the
bob is a box of sand with a slow leak. It also "proves" that you can't
increase the energy of a swingset (at constant ω) by slow parametric
pumping.

And that page is just one example among many. The literature is crawling
with stuff like this ... stuff that doesn't pass the giggle test.

There are two ways of using the literature:
*) One way is to study the literature and understand it. That includes
checking that the first principles and the conclusions and everything
in between are consistent with everything else you know. This is the
right way to do it.
*) Another way is to quote some result from the literature without
understanding the result or where it came from. I don't recommend
this approach. You can't believe everything you read.

On 06/09/2011 06:40 AM, Moses Fayngold wrote:
An example with a person periodically bending on a swing is also somewhat
tricky.

It's not any trickier -- or less tricky -- than any other elementary physics
problem. By way of analogy, suppose we say "the swamp boat accelerates ...."
As always, certain parts of the answer depend on what we consider "the system"
and where we draw the system boundary. In some sense it matters whether the
boat accelerates because the natural wind is blowing against it, or whether
it accelerates because the on-board propeller is producing thrust. OTOH the
two analyses have a great deal in common, and indeed if we draw the system
boundary such that it cuts across the pylon that supports the thruster, then
the thruster is no longer "on board" and the two analysis become the same for
all practical purposes.

The same reasoning applies to the pendulum. The original question asked about
"shortening the rod" without specifying whether the shortening mechanism was
internal to "the system" or not. This is just one of many ways in which the
original question was underspecified.

Specifically: Pumping a pendulum by flexing and extending your legs differs
in only the most superficial ways from pumping a pendulum by "externally"
pulling on the string as shown in this diagram:
http://www.av8n.com/physics/img48/dpa.gif


On 06/08/2011 04:44 PM, curtis osterhoudt wrote:
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.

Suppose the length of the pendulum varies as
L = L0 + λ sin( ωt + φ ) [1]
for some λ very small compared to L0, and for some phase φ.

It appears we are being told that if we are ignorant of φ then equation [1]
is twice differentiable ... but if we know φ it is not. Really???!?

See also next message.