10182

# Ring of Pendulums

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.

### DETAILS

The sine-Gordon equation can be written as
with the initial condition . The differential equation must be solved numerically.
References
[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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.