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


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

for all admissable real functions $h$.

By lemma 1 of Legendre's Condition,


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

where


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

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