Introduction
We consider the linear chain of N pendulums each coupled to the nearest neighbors by the elastic bonds. Let us denote the angle of rotation of -th pendulum by and the angle velocity by . Each pendulum experiences the angular moment due to the gravity and the angular moments from the two elastic bonds, which are proportional to the difference in angles of rotation of the coupled pendulums with the coefficient . Under the assumption that all pendulums have the same moment of inertia, J, the set of equations of motion takes the form Ref. [1]
where is the angular moments from the elastic bonds and is the gravity angular moment.
The angular moments from the left and right elastic bonds are and , respectively and hence
The gravity angular moment can be expressed as
> | Gamma2[i]:=-M*d*g*sin(phi[i]); |
where is the gravity constant.
In view of the expressions for and , the equations of motion can be rewritten as
> | J*diff(phi[i](t),t$2)=kappa*(phi[i+1](t)-2*phi[i](t)+phi[i-1](t))-M*d*g*sin(phi[i](t)); |
The above set of equations of motion transforms in the continuous limit to the following partial differential equation
> | J*diff(phi(x,t),t$2)=Kappa*diff(phi(x,t),x$2)-K[G]*sin(phi(x,t)); |
where , K= h ^2.
In the new spatial and time variables, , , the last equation obtains the standard form of the sine-Gordon equation
One can see that the system of coupled pendulums is described by the discrete analog to the sine-Gordon equation. In the following, we will demonstrate some of the solutions to the sine-Gordon equation, using the model described above.