I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically equation $8$.
There is a functional $F$ and we want to find a function $f$ which minimizes $F$. Then we attain the Euler-Lagrange function $E-L(f)$. And we iterate by $\frac{\delta f}{\delta t} = -(E - L)$ to get this $f$.
Why do we set up the problem as $\frac{\delta f}{\delta t} = -(E - L)$ and how we choose the $\delta t$?. The $f$ which meets the $E-L=0$ is a nextreme. Although 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?