Definition:Conjugate Point/wrt original Functional

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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 ) $.