Order and Chaos in Multiple Pendulums

Both plural forms of pendulum - pendula and pendulums - are valid. The former is judged rare and archaic, although it is said to be prevalent in physics. As a physical phenomenon, pendulum - a weight on a string - obeys a non-linear equation. When the non-linearity is assumed small and is abstracted out, the motion is harmonic and satisfies a second degree linear differential equation.

The period of the harmonic motion of a pendulum depends on the length of the string but not on the weight. The applet below shows a number of dots involved in harmonic motion with different periods. The periods are successive integers starting with the one specified by the "Start with" control. The motion could be instantaneously changed from forward (>) to backward (<) and vice versa; it coould also be stopped (||) and again restarted (<,>).

What is attractive in the simulated phenomenon is the interlace between the order and chaos: some of the time the combined motion of several pendulums appear chaotic and then an order emerges and, for a while the pendulums seem to move in concert until the order breaks down and becomes chaotic, only to reemerge a moment later.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

There are several implementations of the pendulums experiment on the web and youTube:

  1. Pendulum Waves (video)
  2. Mr. Honner's (GeoGebra)

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