With Piwik, I noticed that Derivation of the Euler-Lagrange-Equation is one of the most popular pages on my site. During the physik441 Numerische Methoden der Physik course, I created an animation for the double pendulum. Using that source code, I went through some other simple mechanical problems that were covered in my physik221 Theoretische Physik 1 lecture and made animations for those. The derivation of the differential equations are on the pages as well.
There are the following systems:
You can find the source code for both the numerical integration of the differential equations, as well as the animation itself, on the project page: lagrange-simulator.