Jacobi's Necessary Condition/Dependent on N Functions

Theorem
Let $ J $ be a functional, such that:


 * $ J \left [ { \mathbf y } \right ] = \displaystyle \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 real vector.

Let $ \mathbf y \left ( { x } \right ) $ correspond to the minimum of $ J $.

Let the $ N \times N $ matrix $ \mathbf P = F_{ y_i' y_j' } $ be positive definite along $ \mathbf y \left ( { x } \right ) $.

Then the open interval $ \left ( { a \,. \,. \, b } \right )$ contains no points conjugate to $ a $.

Proof
By Necessary Condition for Twice Differentiable N Function dependent Functional to have Minimum, $ J $ is minimised by $ y = \mathbf{ \hat y } \left ( { x } \right ) $ if


 * $ \displaystyle \delta^2 J \left [ { \mathbf { \hat y }; \mathbf h } \right ] \ge 0 $

for all admissable real functions $ \mathbf h $.

By lemma 1 of Legendre's Condition,


 * $ \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


 * $ \mathbf P = F_{ y_i' y_j' } $

By Nonnegative Quadratic N function dependent Functional implies no Interior Conjugate Points, $ \left ( { a \,. \,. \, b } \right )$ does not contain any conjugate points $ J $.