The elastic ribbon sine-Gordon model

[standardflop]

Hello,
I'd like to verify that the elastic ribbon model [ depicted here: http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/sg-e.html ] is governed by the sine-Gordon equation. I suppose this can be shown by writing the lagrangian $L = T - V$ and looking at the variation. The kinetic energy for a single single pendulum is $T =\tfrac{1}{2} \dot{\phi}^2$, but how can i describe potential $V$ now that each pendulum is coupled to its neighbours?

All the best
SF

 

[standardflop]

According to Wikipedia the the s-G equation is the Euler-Lagrange equation of the following lagrangian
$$\mathcal{L}(\phi) = \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.$$
Thus i suppose my question is simply how to derive this lagrangian for the mentioned mechanical system.