Jacobi's Necessary Condition

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


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

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

Let:


 * $F_{y'y'}>0$

along $\map y x$.

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

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


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

for all admissable real functions $h$.

By lemma 1 of Legendre's Condition,


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

where:


 * $P = F_{y' y'}$

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