Legrangian problem types

[ChrisRackauckas]

The Lagrangian L(t,x,dx) can be used to specify the equations of motion as well. My recall of this approach is quite hazy (I have always used Hamiltonians: is there a field where using Lagrangians directly is useful?). I'm not entirely sure how to do this without getting an implicit equation for the second derivative (this is easy to solve for usually symbolically, but how would it be done numerically?)