= Jacobi positivity criterion
{c}
{title2=$q>0\iff\text{no conjugate point in }(a,b]$}
Assume the <Strengthened Legendre condition> $P>0$ and let $q(\varphi)=\int_a^b(P\varphi'^2+Q\varphi^2)$. Then $q$ is strictly positive on all nonzero fixed-endpoint <Lipschitz> variations exactly when the normalized <Jacobi equation> solution with $w(a)=0$, $P(a)w'(a)=1$ has no zero in $(a,b]$. A zero at $c$ gives a nonzero variation equal to $w$ up to $c$ and zero thereafter, with $q=0$. Conversely, if no zero occurs, the <Jacobi identity for the second variation> writes $q$ as a nonnegative square. Equality forces $\varphi=Cw$, and the endpoint at $b$ forces $C=0$.
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