Definition:Twice Differentiable/Functional

Definition
Let $ \Delta J \left [ { y; h } \right ]$ be an increment of functional

Let $\Delta J \left [ {y; h } \right]= \phi_1[y; h]+ \phi_2[y; h]+\epsilon \left\vert h\right\vert^2$,

where $ \phi_1 \left [ {y; h } \right ]$ is a linear functional and $ \phi_2 \left [ { y; h } \right ]$ is a linear functional $ h$, and $ \epsilon \to 0$ as $ \left \vert h \right \vert\ to $ 0.

Then the functional $J \left [ {y } \right ]$ is said to be twice differentiable.

The linear part $ \phi_1$ is the first variation, denoted by $ \delta J \left [ {y; h } \right ]$.

$ \phi_2$ is called the second variation (or differential) of a functional, and is denoted by $\delta J^2 \left [ { y; h } \right ]$.