How is the Euler-Lagrange equation:
$$ L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 $$
used mathematically in finding the optimal solutions of minimising a function? Can someone give me an example please?
I'm trying to determine how this formula is applicable to the study of minimal surfaces?