 Euler-Lagrange equation for a second-derivative functional

For a sufficiently smooth functional $I[y]={\int }_a^{b}F(x,y,y^{\prime },y^{\prime \prime })dx$, the first variation in direction $\eta$ is $\int (F_y\eta +F_{y^{\prime }}{\eta }^{\prime }+F_{y^{\prime \prime }}{\eta }^{\prime \prime })dx$. Twice applying integration by parts gives the interior coefficient $F_y-(F_{y^{\prime }}{)}^{\prime }+(F_{y^{\prime \prime }}{)}^{\prime \prime }$ and boundary expression $[(F_{y^{\prime }}-(F_{y^{\prime \prime }}{)}^{\prime })\eta +F_{y^{\prime \prime }}{\eta }^{\prime }{]}_a^b$. 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.