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

Lemma
Let $ \mathbf y = \left ( { \langle y_i \left ( { x } \right ) \rangle_{1 \le i \le N} } \right ) $ be a vector real function, such that:


 * $ \mathbf y \left ( { a } \right )= A, \quad \mathbf y \left ( { b } \right ) = B$

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


 * $ F \in C^2 \left [ { a \,. \,. \, b } \right ] $

all its variables.

Then


 * $ \displaystyle \delta^2 J \left [ { \mathbf y; \mathbf h } \right ] = \int_a^b \left ( { \mathbf h' \mathbf P \mathbf h'+ \mathbf h \mathbf Q \mathbf h } \right ) \mathrm d x$

where


 * $ \displaystyle \mathbf P = \frac{ 1 }{ 2 } F_{ y_i' y_j' }, \quad \mathbf Q = \frac{ 1 }{ 2 } \left ( { F_{ y_i y_j} - \frac{ \mathrm d }{ \mathrm d x } F_{ y_i y_j'} } \right )$