Legendre's Condition/Lemma 1/Dependent on N Functions

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Lemma

Let $\mathbf y = \paren {\sequence {\map {y_i} x}_{1 \mathop \le i \mathop \le N} }$ be a vector real function, such that:

$\map {\mathbf y} a = A$

$\map {\mathbf y} b = B$

Let $J \sqbrk {\mathbf y}$ be a functional, such that:

$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$

where:

$F \in C^2 \closedint a b$

with respect to all its variables.

Then:

$\ds \delta^2 J \sqbrk {\mathbf y; \mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$

where:

$\mathbf P = \dfrac 1 2 F_{y_i'y_j'}$

$\mathbf Q = \dfrac 1 2 \paren {F_{ y_i y_j} - \dfrac \d {\d x} F_{y_i y_j'} }$

Proof

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Sources

1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.29$: Generalization to n Unknown Functions