Definition:Twice Differentiable/Functional

Definition

Let $\Delta J \sqbrk {y; h}$ be an increment of a functional.

Let:

$\Delta J \sqbrk {y; h} = \phi_1 \sqbrk {y; h} + \phi_2 \sqbrk {y; h} + \epsilon \size h^2$

where:

$\phi_1 \sqbrk {y; h}$ is a linear functional

$\phi_2 \sqbrk {y; h}$ is a quadratic functional with respect to $h$

$\epsilon \to 0$ as $\size h \to 0$.

Then the functional $J\sqbrk y$ is twice differentiable.

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The linear part $\phi_1$ is the first variation, denoted:

$\delta J \sqbrk {y; h}$

$\phi_2$ is called the second variation (or differential) of a functional, and is denoted:

$\delta^2 J \sqbrk {y; h}$

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.24$: Quadratic Functionals. The Second Variation of a Functional