The Double Pendulum

[Animated gif (109kB)]
Animated gif (109kB) showing solution of the double pendulum equations for particular initial conditions.

[Animated gif (239kB)]
Animated gif (239kB) showing two solutions of the double pendulum equations for slightly differing initial conditions.

The double pendulum is an example of a simple dynamical system that exhibits complex behaviour, including chaos. It consists of two point masses at the end of light rods. Each mass plus rod is a regular simple pendulum, and the two pendula are joined together and the system is free to oscillate in a plane.

This page has an excellent, detailed description of the dynamical description of the double pendulum, including derivation of the equations of motion in the Lagrangian formalism. For the purposes of numerical solution the equations of motion may be written in the standard form

The left panel shows two animated gifs illustrating solution of these equations for one kilogram pendulum masses and one metre pendulum lengths, for the indicated times in seconds (the two gifs may take some time to load - they are 109kB and 239kB respectively). The second gif illustrates sensitivity to initial conditions: the pendulum is started from two initial configurations differing by 0.001 degrees in the angle of the second mass.

The C code used to solve the equations is here. Fourth order Runge-Kutta is used to integrate the ODEs. It should be noted that this approach does not give accurate solutions for long times for equations such as these - symplectic integrators should be used. (Or so the applied mathematicians tell me.)

The School of Physics at the University of Sydney has a compound square double pendulum.

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