Some time ago, before I had a Mathematica license, I solved the classical pendulum problem with only algebra and rational numbers :

http://grondilu.github.io/pendulum.html

I was inspired by Norman Wildberger who advocates on YouTube for what he calls rational trigonometry.

I wanted to do the same for the double pendulum, but the maths were too intimidating. With Mathematica doing all the tedious calculus part, I finally managed to do it. I'm happy with it, even though I can't yet deal with an arm doing a complete turn, because of the singularity in the stereographic mapping.

I'm planning on fixing this by changing the local map at each time increment. That's why I made sure the parametrization of the configuration space can be defined in any neighborhood. I will try to make the map transfer function work later, but I'm already satisfied enough with this little project to share it.

PS: the graphics do not seem to show up in the cloud version. Not sure why.

PS#2: I managed to export the Euler-Lagrange equations to javascript so I could make a web, real time, continuous simulation : http://grondilu.github.io/double-pendulum.html. I'm not quite sure how I could do that in Mathematica yet, hopefully I will someday.