Ring of Pendulums

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Consider a large collection of coupled pendulums of the same length. Passing to the continuum limit, these can be solved by means of the sine-Gordon equation with periodic boundary conditions; the pendulums can be thought of as attached to a ring. The solutions as a function of time are shown with two initial peaks separated by a distance . Soliton-like behavior can be observed.

Contributed by: Enrique Zeleny (July 2014)
Open content licensed under CC BY-NC-SA



The sine-Gordon equation can be written as

with the initial condition . The differential equation must be solved numerically.


[1] J. V. José snd E. J. Saletan, Classical Dynamics: A Contemporary Approach, 2nd ed., Cambridge: Cambridge University Press, 2013.

[2] U. Pinkall. Pendulum Chain, Part 1 [Video]. (Apr 20, 2011) www.youtube.com/watch?v=Ozt1VkeK52E.

[3] U. Pinkall. Pendulum Chain, Part 2 [Video]. (Apr 21, 2011) www.youtube.com/watch?v=iOc3idE88LM.

[4] J. Cuevas-Maraver, P. Kevrekidis, and F. Williams, eds., The Sine-Gordon Model and Its Applications, New York: Springer, 2014.

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