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

From ProofWiki
Jump to: navigation, search

Lemma

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 $\mathbf y=\paren {\sequence {y_i}_{1\le i\le N} }$ 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:

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

Let

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


Then $\mathbf P$ is nonnegative.


Proof


Sources

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