How to find the Euler Lagrange Equation of this energy function?
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I come across this problem: Find the Euler Lagrange Equation of this energy function begin{equation*} J(u) = int_{0}^{1} c(x) left(frac{mathrm{d}u(x)}{mathrm{d}x}right)^{2} mathrm{d}x end{equation*} So $frac{mathrm{d}J(u + tv)}{mathrm{d}t}$ at $t = 0$ would be $2 int_{0}^{1} c(x) left( frac{mathrm{d}u(x)}{mathrm{d}x} right)left( frac{mathrm{d}v(x)}{mathrm{d}x}right) mathrm{d}x$ . However, without the differentiability of $c(x)$ , I cannot do the integration by part, choose $v(x)$ that is compactly supported inside $(0,1)$ and obtain the differential equation. Do you think that it is just the problem forgot to mention the differentiability of $c(x)$ , or it is in fact not necessary?
calculus
