I'm self teaching myself calculus of variations, and decided to solve a problem to practice what I learned. Say we want to find a path $y=y(x)$ in the scalar field $S(x,y)$. Therefor we wish to minimise the integral $$\int_{x_1}^{x_2} S(x,y(x)) dx $$ Let's consider a seamingly simple example where $S(x,y)=y*x^2$ $$\int_{x_1}^{x_2} y*x^2 dx $$ Applying the Euler-Lagrange equation $$x^2-\frac{d}{dx} 0=0$$ Therefore $x^2=0$ Which just leaves me confused, as it don't give us any information about $y(x)$ Did I do something wrong or is this a limitation of the Euler-Lagrange? Is there a way to solve such problems?
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In your example there isn't maximum nor minimum. Let $y(x)=c$ constant $$\int_{x_1}^{x_2} yx^2 dx=c\int_{x_1}^{x_2} x^2 dx$$ and take $c$ big positive or big negative.