The second problem is with the lower angle (θ 2 ). This angle is measured from a vertical line but this variable by itself does not give the whole motion of the lower mass. Angle θ 2 could stay at zero but the lower mass could still be moving due to the motion of mass 1. This means that the time derivatives of θ 2 can be rather complicated.

In the end the best method to solve this problem is to use Lagrangian mechanics—a system that uses energy and constraints to obtain an equation of motion. For the double pendulum, Lagrangian mechanics can get an expression for angular acceleration for both angles (the second derivative with respect to time) but these angular accelerations are functions of both the angles and the angular velocities. There is no simple solution for the motion of the two masses. Really, you need to do a numerical calculation using some type of computer code to find the motion of the system.

If you want to go over all the details of getting a double pendulum solution, check out this site—it does a fairly nice job showing how to get expressions for the angular accelerations.

For my model, I am going to use Python (hopefully, you could have guessed that). Here is what I get. Just a note, you can look at and change the code. But first, just run it by pressing "play" to run and "pencil" to edit. If the model stops running, just click the "play" button again to start over.

I put some comments at the top of the code to point out the things that you might want to change. The first thing to try is starting with different initial angles of θ 1 and θ 2 —but you can also change the value of the masses and the lengths of the strings. It's pretty fun to watch it move around.

Chaotic System

The double pendulum is a great example of a chaotic system. What does that even mean? Let me start with an example. Here are two double pendulums right on top of each other (well, almost). For one of the pendulums the starting angle for the lower mass is just 0.01 degrees different than the other pendulum—so they essentially start with the same initial conditions. Watch what happens as the two double pendulums swing back and forth. Again, you can click "play" to run it more than once.