Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] Classical Adiabatic Invariant

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)}.


 
/**************************************
"The four points of the compass be logic, knowledge, wisdom and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three. I may submit to the unknown, but never to the unknowable." ~~Roger Zelazny, in "Lord of Light"
***************************************/


________________________________
From: John Denker <jsd@av8n.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Wednesday, June 8, 2011 12:00 PM
Subject: Re: [Phys-l] Classical Adiabatic Invariant

On 06/06/2011 10:53 AM, Bernard Cleyet wrote:
I'm attempting to understand this as applied to a simple pendulum.
The examples I've  found apply to shortening the rod.  Wherein
E/omega is a constant.  Shortening the rod increases the frequency,
so the E must increase.

On 06/06/2011 06:26 PM, we got a clarification:
The changes are very slow.  i.e.  The increase  of v. ~ 60
microgals, (in ~9.8E+8micro. ) which "takes" ~12 hours.  The period
of the pendulum is 2Hz.

In the limit of very slow changes, this is an easy problem.
It involves nothing more than high-school physics.

If you approach it the right way, you should be able to do
it in your head, in less time than it takes to tell about it.

Hint:  The energy is *not* proportional to omega.

Bigger hint:  There are some classical invariants that can
be invoked.  E/omega is just not one of them.

Ginormous hint:
Ercynpr gur ebq ol n fgevat gung cnffrf guebhtu n ubyr va
na vzzbinoyr gbc-cyngr.  Chyyvat ba gur fgevat unf gur fnzr
arg rssrpg nf fubegravat gur ebq ... ohg vg znl or rnfvre
gb nppbhag sbe gur sbepr ba gur fgevat.
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l