The elastic ribbon sine-Gordon model

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SUMMARY

The elastic ribbon model is governed by the sine-Gordon equation, as established through the Lagrangian formulation. The kinetic energy for a single pendulum is defined as T = \tfrac{1}{2} \dot{\phi}^2. The challenge lies in accurately describing the potential energy V, considering the coupling between neighboring pendulums. To derive the Lagrangian for this mechanical system, one must apply the principles of the Euler-Lagrange equation, specifically using the form \mathcal{L}(\phi) = \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with the sine-Gordon equation
  • Knowledge of kinetic and potential energy in mechanical systems
  • Basic grasp of coupled oscillators
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  • Study the derivation of the sine-Gordon equation from Lagrangian mechanics
  • Explore the concept of coupled pendulums in classical mechanics
  • Investigate the applications of the sine-Gordon equation in soliton theory
  • Learn about the Euler-Lagrange equation and its applications in physics
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Physicists, mechanical engineers, and researchers in nonlinear dynamics who are interested in the mathematical modeling of coupled oscillatory systems.

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 [itex]L = T - V[/itex] and looking at the variation. The kinetic energy for a single single pendulum is [itex]T =\tfrac{1}{2} \dot{\phi}^2[/itex], but how can i describe potential [itex]V[/itex] 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
[tex]\mathcal{L}(\phi) = \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.[/tex]
Thus i suppose my question is simply how to derive this lagrangian for the mentioned mechanical system.
 

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