I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9...
There is a functional F and we want to find a function f which minimize F. Then we attain the Euler-lagrange function E-L(f). And we iterate by δf/δt=−(E−L) to get this f.
My question is "why do we set up the problem as δf/δt=−(E−L) and how we choose the δt". The f which meets the E-L=0 is a extreme. Althought the gradient descent method is also a iteration processing, the right side of its equation is - $\nabla$f if one want the local minimum. But does E-L is a type of $\nabla$f.
How we choose $\delta$t, if it is too big, the iteration process is not stable, if it is too small, it is time-consuming. There is a necessary condition - CFL condition, not sufficient condition .Also, this condition is decided by the right side of the equation. link In page three, part 3, $\delta$t is decided by the order spatial derivatives. But in my equation, I do not have any spatial derivatives,but I have a integral, $\int$f, So how should I choose the $\delta$t, I ask a professor in my uni,the answer is 'It is not a analytical function,you need to try the number'.
PS: Could you understand my question?