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

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Lemma

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

$\map {\mathbf y} a=A,\quad \map {\mathbf y} b=B$

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

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

where

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

with respect to all its variables.


Then

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

where

$\displaystyle\mathbf P=\frac 1 2 F_{y_i'y_j'},\quad\mathbf Q=\frac 1 2 \paren { F_{ y_i y_j}-\frac \d {\d x} F_{y_iy_j'} }$


Proof


Sources

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