Let $f_{i} \colon [0,L] \to \mathbb{R}$ be three times differentiable. I need to consider a functional $S$ of the following form: \begin{equation*} S(f_{1},f_{2},f_{3}) = \int_{0}^{L}L(t, f_{1},f_{2},f_{3},f_{1}',f_{2}',f_{3}',f_{1}'',f_{2}'',f_{3}'',f_{1}''',f_{2}''',f_{3}''')\,dt, \end{equation*} where an apostrophe denotes differentiation with respect to the variable $t$.
I would like to write down the corresponding Euler–Lagrange equation(s). I tried to derive the result from the more general case treated on Wikipedia, but I got rather confused by the index notation. Can anybody help?
Applying single function of single variable with higher derivatives case from the page to each function $f_i$ separately and you get $$ \frac{\partial L}{\partial f_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial f_i'}+\frac{\mathrm{d}^2}{\mathrm{d}t^2}\frac{\partial L}{\partial f_i''}-\frac{\mathrm{d}^3}{\mathrm{d}t^3}\frac{\partial L}{\partial f_i'''}=0. $$