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.

 

[ogb p]

Hello SF,

Thank you for your question. The elastic ribbon model is indeed governed by the sine-Gordon equation, which is a nonlinear partial differential equation that describes the motion of a classical field in one spatial dimension. The equation can be derived from the energy functional, which is the integral of the Lagrangian over the space and time.

In the case of the elastic ribbon model, the Lagrangian can be written as L = T - V, where T is the kinetic energy and V is the potential energy. The kinetic energy is given by T = \tfrac{1}{2} \dot{\phi}^2, as you mentioned. However, in order to describe the potential energy, we need to consider the interactions between neighboring pendulums.

The potential energy in the elastic ribbon model can be described as a sum of potential energies between each pair of neighboring pendulums. This can be written as V = \sum_{i=1}^N V_i, where N is the number of pendulums and V_i is the potential energy between the i-th and (i+1)-th pendulums. This potential energy can be further expanded using Hooke's law, which describes the restoring force of a spring, to include the elastic properties of the ribbon.

By considering the kinetic and potential energies, and using the Euler-Lagrange equation, we can derive the sine-Gordon equation to describe the dynamics of the elastic ribbon model. This equation takes into account the coupling between neighboring pendulums and the elastic properties of the ribbon.

I hope this helps to clarify the connection between the elastic ribbon model and the sine-Gordon equation. If you have any further questions, please don't hesitate to ask.

Best regards,