Legendre's Condition/Lemma 2/Dependent on N Functions

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Lemma

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 $\mathbf y = \paren {\sequence {y_i}_{1 \mathop \le i \mathop \le N} }$ is an N-dimensional vector.

Let $F \in C^2 \closedint a b$ with respect to all its variables.

Let $\mathbf P$, $\mathbf Q$ be $N \times N$ real matrices, such that:

$\mathbf P = \dfrac 1 2 \dfrac {\partial^2 F} {\partial y_i'\partial y_j'}$

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

Let:

$\ds \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x \ge 0$

Then $\mathbf P$ is nonnegative.

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