= Euler-Lagrange equation for a second-derivative functional
{c}
{title2=$F_y-(F_{y'})'+(F_{y''})''=0$}
For a sufficiently smooth <functional> $I[y]=\int_a^b F(x,y,y',y'')\,dx$, the <first variation> in direction $\eta$ is $\int(F_y\eta+F_{y'}\eta'+F_{y''}\eta'')dx$. Twice applying <integration by parts> gives the interior coefficient $F_y-(F_{y'})'+(F_{y''})''$ and boundary expression $[(F_{y'}-(F_{y''})')\eta+F_{y''}\eta']_a^b$. Variations supported in the interior and the <fundamental lemma of the calculus of variations> force the displayed equation. The remaining boundary terms give the <natural boundary conditions for second-derivative functionals>.
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