Definition:Conjugate Point (Calculus of Variations)/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 $ to the original functional $ \displaystyle \int_a^b F \left ( { x, y, y' } \right ) $.