# Definition:Conjugate Point/wrt original Functional

## Definition

Let

$\displaystyle \int_a^b F \left ( { x, y, y' } \right )$

be the original functional.

Let $\tilde a$ be conjugate to $a$.

Let

$\displaystyle \displaystyle \int_a^b \left ( { Ph'^2 + Qh^2 } \right ) \mathrm d x$

be the second variation of $\displaystyle \int_a^b F \left ( { x, y, y' } \right )$.

Then $\tilde a$ is conjugate to $a$ with respect to to the original functional $\displaystyle \int_a^b F \left ( { x, y, y' } \right )$.