I don't know if it's the right place, but I found a link who derive the lagrange's equation without the use of the variational calculus or the action functional. I thought it would be cool for highschoolers and college students who are eager to have a glimps of what is lagrange equation! Not formal nor complete, but still nice.
Enjoy!
http://http://www.eftaylor.com/pub/lagrange.html 
