# 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 ) $.