Date: Mon, 18 Oct 2004 13:44:03 -0400

Author: Andrew Graham

Subject: Re: localized energy mode demo, i.e pendulum revisited


Andrew Dougherty wrote:

>On Mon, 18 Oct 2004, Andrew Graham wrote:
>>Thanks for sharing this. The physical model you describe is the basis
>>for the famous Sine-Gordon wave equation. This is closely akin to the
>>area of theoretical physics that I work in. The kink-antikink solutions
>>are well studied, and represent the "particle-antiparticle" pair created
>>when one pendulum goes through a full rotation and back to the
>>equilibrium position, producing two stationary twists along the row of
>>pendula. ( I have a similar demo that I copied from one made by Dick
>>Berg.) However, the solution you describe is a small oscillation effect
>>and very interesting. Do you have a reference to a publication
>>discussing this effect? There is probably an analogous solution in the
>>system I study.
>Offhand, it sounds like an example of FPU (Fermi, Pasta, and
>Ulam) Recurrence. There's a nice simulation of this in the Wiley CUPS
>series "Waves and Optics Simulations" by Antonelli, Christian, Fischer,
>Giles, James, and Stoner. There are probably nice Java applets posted
>somewhere on the web to do the same thing these days.

Thanks for that clarification. You are correct that the apparatus Sam
described is an FPU model. I should have said that this is the discrete
model that gives, in the continuum limit (as the separation between
adjacent pendula goes to zero), the Sine-Gordon wave equation. My
research begins with a discrete linear lattice of masses connected with
springs, and in the continuum limit gives the Klein-Gordon wave equation.

andy graham
From Mon Oct 18 11:22:49 2004