If you read some introductory mechanics text like David Morin's Introduction to Classical Mechanics about Euler Lagrange Equations you get a large amount of simple examples like the "moving plane" (Problem 6.1 in the link above) or the double pendulum of how to apply the Euler Lagrange equations.
However something like the moving plane example or the double pendulum looks for me like a nice toy example.
Are any of those typical simple examples relevant in modelling something in engineering, sports science or something like that? If not are there any simple examples for the applications of the Euler Lagrange equations which are relevant in this sense?
By "simple" I mean something like the double pendulum or even more simple (in deriving the equations of motion; not solving it); the more simple the example the better.
P.S. I am not sure it the question is more appropriate on engineering.sx. But I cannot crosspost. If you think it should be migrated, please let my know.