Euler-Lagrange

The general equation for the case where you're dealing with

$\mathcal{L}(x, f', f'', \dots, f^{(N)})$

is given by

$\displaystyle \partial_f{\mathcal{L}} + \sum_{k=1}^{N} \left(-1\right)^{k}\frac{d^k}{dx^k}\left(\partial_{f^{(k)}}\mathcal{L}\right) = 0$

where N is the highest order of the derivative inside your Lagrangian. In your case, you'd just be dealing with $N=2$ and hence you'd end up with

$\frac{\partial{\mathcal{L}}}{\partial{f}} - \frac{d}{dx}\left(\frac{\partial\mathcal{L}}{\partial{f'}}\right) + \frac{d^2}{dx^2}\left(\frac{\partial\mathcal{L}}{\partial{f''}}\right) = 0[br]$

Can you suggest a book for learning the calculus of variations (not for absolute beginners)?

Honestly I just use Wikipedia, either that or the course notes they give us here at Uni. To supplement both, "Mathematical Methods for Physics and Engineering" is a good book, that big thick one that costs like £40... by Riley/Hobson or whatever their name is.