I am new to calculus of variation and try to understand some simple special cases. In particular, what is the second variation for a functional
$$J[y]=\int_{a}^{b} F(x, y(x)) dx$$
without the $y'(x)$ term in $F$? Is it
$$
\frac{1}{2}\int_{a}^{b} \left[\frac{\partial^2}{\partial y^2} F(x,y(x))\right] \cdot \left[y(x)\right]^2 dx
$$
or am I missing something trivial? I got the above formula from Section 5 in Gelfand and Fomin's textbook (see screenshot attached) and simply take out the second and third terms inside the integral in Equation (9). 
Any hint is highly appreciated! Thanks in advance.
Best Answer
Premise: this MathOverflow Q&A on functional derivatives on Banach spaces and this one on higher order functional derivatives may be useful.
Answer. Second and higher order variations may be defined in the most general way as follows $$ \begin{split} \frac{\delta^n}{\delta h_1 \delta h_2\cdots\delta h_n}J[y] &\triangleq \frac{\partial^n}{\partial \varepsilon_1 \partial \varepsilon_2\cdots\partial \varepsilon_n}J[y+\varepsilon_1 h_1 + \varepsilon_2 h_2+\ldots + \varepsilon_n h_n]\bigg|_{\varepsilon_i =0\; i=1,\ldots,n}. \end{split} $$ If we can choose $h_i =h $ and $\varepsilon_i =\varepsilon$ for all $i = 1,\ldots, n$ we have the following simplified formulation: $$ \begin{split} \frac{\delta^n}{\delta h^n }J[y] &\triangleq \frac{\partial^n}{\partial \varepsilon^n }J[y+\varepsilon h]\bigg|_{\varepsilon = 0}\\ \end{split} $$ Be it noted that the second form of $n$-variation is particularly suited to deal with integral type functionals $J$ as it happens in the case under examination: the former form is however more general in that it explicitly show the multilinearity of the higher variation functional and it is thus applicable to the case where the variations $h_1, \ldots, h_n$ are not functions but more generally generalized functions.
For $n=2$, by applying the latter definition to the very special form of the functional $J[y]$ at hand, we have $$ \frac{\delta^2}{\delta h^2 }J[y](h) = \int\limits_a^b \left[\frac{\partial^2}{\partial y^2} F(x,y)\right] |h(x)|^2\operatorname{d\!}{x} $$ Note that including the factor $1/2$ in the definition of the second variation is a notational convention adopted by Gel'fand and Fomin in their textbook: Giaquinta and Hildebrandt do not follow this convention in their two volume monograph.