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

Lemma
Let $ J \left [ { \mathbf y } \right ] $ be a functional, such that:


 * $ \displaystyle J \left [ { \mathbf y } \right ] = \int_a^b F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x $

where $ \mathbf y = \left ( { \langle y_i \rangle_{ 1 \le i \le N } } \right ) $ is an N-dimensional vector.

Let $ F \in C^2 \left [ { a \,. \,. \, b } \right ] $ 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 } \left ( { \frac{ \partial^2 F }{ \partial y_i \partial y_j } - \frac{ \mathrm d }{ \mathrm d x } \frac{ \partial^2 F }{ \partial y_i \partial y_j'} } \right )$

Let


 * $ \displaystyle \int_a^b \left ( { \mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h } \right ) \mathrm d x \ge 0 $

Then $ \mathbf P $ is nonnegative.