This is a continuation of this question whose general point is summarised below
Say we want to find a path $y=y(x)$ in the scalar field $S(x,y)$ that finds the extrema of of its line integral. Therefore we wish to minimise the integral $$\int_{x_1}^{x_2} S(x,y(x)) dx $$
The original question involved $S(x,y)=y*x^2$ and the Euler-Lagrange equation returned $x^2=0$ because there was no extrema.
But intuitively $S(x,y)=x^2*y^2$ should have a minimum path for almost any two point. Applying the Euler-Lagrange: $$\frac{dS}{dy}-\frac{d}{dx}(\frac{dS}{dx'})=0 $$ $$2*x^2*y-\frac{d}{dx}0=0 $$ $$x^2*y=0 $$ Which means either $x=0$ or $y(x)=0$ which continues to confuse me, as I assumed a minimum would exist. Could somebody please explain to my how the Euler-Lagrange equation behaves for the integrand being independent of the derivative of $y$?