I was going through the Ritz method in Finite Element analysis when I came across the statement here (see equation 1.25, its not same but similar. below I have taken a problem from my book.),
Consider a Boundary Value Problem in 1-D having a differential equation,$$ \frac{d^2f}{dx^2} =-g(x)$$ for $$a<x<b$$ The functional equivalent of the above D.E. is written as $$J(f)=\int_a^b \left(\frac{1}{2}\left[\frac{df}{dx}\right]^2-f(x)g(x)\right)dx$$Equation above is obtained by intuition, but it is shown to really give the functional of the first equation. So, the solution to the differential equation is equivalent to minimizing the above quadratic functional.
What is that intuition that gives such a functional relation for f? Isn't there a definite way to obtain such an integral statement? Can anyone help me with the steps leading this DE to this integral statement?
Without fundamentals, according to the Euler-Lagrange criteria, the stationary points to the functional
$$ J(f) = \int_a^b L(f',f,x)dx $$
are obtained by solving
$$ \frac{d}{dx}\left(\frac{\partial L}{\partial f'}\right)-\frac{\partial L}{\partial f}=0 $$
now calling $L = \frac 12\left(f'\right)^2-f g$ we are done. If the functional $J(f)$ has a minimum, then the Euler-Lagrange criteria gives it.