In physics people do the variation of a quantity typically of the form $$ \delta_X f(X,Y,Z)=0 $$ where in general the variation $$X\rightarrow X+\delta X $$ where $$\delta X\propto \epsilon g(X) $$ of some small variable $\epsilon$ and only the first order variation was considered. However, it seemed that the quality of the interest through this procedure were mostly first order, by which I mean at most a line integral $$\int dX \text{(the quantity of the interest)} $$ or if a transformation was required $$H(X,Y,Z)\rightarrow H(\tilde X, Y,Z) +\Delta X T(X,Y,Z)$$
However, what if the quantity of the interest was of the second order in nature? i.e. $$f(X,Y,Z)=\int dX \int dX... $$ and $$\int dX \int dX \text{(the quantity of the interest)}$$
Would the second order variation necessary to be kept under this situation?