Consider a regular curve and the functional $$E(\gamma)=\int_{\gamma} k^2 ds,$$ where $k$ denotes the curvature and $s$ the arclength parameter.
I have to compute the first variation. Hence I define $\gamma_\beta=\gamma+\beta\psi $ perturbation of $\gamma$. Now I have to compute $$\frac{\partial}{\partial \beta}(E(\gamma_\beta))=\frac{\partial}{\partial \beta}\int_\gamma k_\beta^2 ds.$$ I also know that $k_\beta=\frac{\partial_{xx}\gamma_\beta}{|\partial_x\gamma_\beta|^2}-\frac{\langle\partial_{xx}\gamma_\beta,\partial_x\gamma_\beta\rangle\partial\gamma_\beta}{|\partial_x\gamma_\beta|^4}$.
I tried to do the math, but without success. Euler-Lagrange equation should come back $2\partial^2_sk+k^3-k=0$. Can you help me or point me to a reference where I can find the steps in detail?