Jacobi's Necessary Condition/Dependent on N Functions

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


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

where $\mathbf y = \paren {\sequence {y_i}_{1 \le i \le N} }$ is an N-dimensional real vector.

Let $\map {\mathbf y} x$ correspond to the minimum of $J$.

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

Then the open interval $\openint a b$ contains no points conjugate to $a$.

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


 * $\delta^2 J \sqbrk {\mathbf {\hat y}; \mathbf h} \ge 0$

for all admissable real functions $\mathbf h$.

By lemma 1 of Legendre's Condition:


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

where:


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

By Nonnegative Quadratic N function dependent Functional implies no Interior Conjugate Points, $\openint a b$ does not contain any conjugate points $J$.