The double pendulum is interesting because it is a simple physical system with surprisingly complex behavior. It shows sensitivity to initial conditions and hence can be considered chaotic for some starting configurations.
Micael Oliveira has started physics-sandbox with the idea of creating some physics demos and the first is the double pendulum which you can see, in a simplified version, below.
From the chaos point of view, what is interesting is how often the bottom pendulum does a complete loop. It is very difficult to predict if or when a loop will occur - so as you watch lookout for a loop about to happen.
If you want to know the theory, go to Micael's web page explaining it. He takes the Lagrangian approach and, after working out the Lagrangian and putting it into the equations of motion, ends up with something that looks horrible. The equations give the angular acceleration as a function of the angular position and velocity.
The power of programming it is that even when you cannot find a closed form for position as a function of time, you can simulate the system. In this case nothing more than the basic Euler method is needed; the new position is the old position plus the relevant velocity multiplied by the time step and the new velocity is the old velocity multiplied by the relevant acceleration multiplied by the time step. This is a direct physical simulation but it seems to be stable enough if the step size is small.