The double pendulum is composed of a second pendulum attached to the end of the bob of an initial simple pendulum, as shown in the diagram below:

Since there are now 2 angles and changing with respect to time t, this system exhibits 2 degrees of freedom.

As a result, the motion of the pendulum is more difficult to model, and requires more complex mathematics than those used to simulate the motion of the Simple Pendulum.

Thus, we now introduce the use of Lagrangian mechanics.

Using Lagrange's equations, we can derive the following equations of motion:
(Click Here to see the proof)

The motion described by the equations above is demonstrated in the interactive applet below:

If you can read this writing then either you do not have Java installed, or your Java is not new enough to view the applets on this site. You need Java 1.4 to view these applets.

In the applet above it is possible to adjust the initial values (at time t=0) of , , and ; as well as , , , and . In addition to this, it is also possible to activate a trace on the second bob in order to make it easier to visualise the motion of the pendulum over time; and you can deactivate gravity to see how the pendulum would react in zero-gravity conditions.

© Michael Hart 2004