The elastic ribbon sine-Gordon model

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The discussion centers on verifying that the elastic ribbon model is governed by the sine-Gordon equation. The user seeks to derive the Lagrangian for the system, starting with the kinetic energy of a single pendulum and questioning how to express the potential energy when pendulums are coupled. The sine-Gordon equation is identified as the Euler-Lagrange equation for a specific Lagrangian involving kinetic and potential energy terms. The user is looking for guidance on how to formulate the potential energy for the coupled pendulum system. The inquiry highlights the need for a clear derivation of the Lagrangian to connect the mechanical system to the sine-Gordon framework.
standardflop
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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
 
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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.
 

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