Definition:Conjugate Point (Calculus of Variations)/Definition 2

Definition
Let $y = \map y x$ and $y^* = \map {y^*} x$ be extremal functions.

Let:


 * $M = \tuple {a, \map y a}$


 * $\tilde M = \tuple {\tilde a, \map y {\tilde a} }$

Let $y$ and $y^*$ both pass through the point $M$.

Let:


 * $\map {y^*} {x - \tilde a} - \map y {x - \tilde a} = \epsilon \size {\map {y^*} {x - \tilde a} - \map y {x - \tilde a} }_1$

where:


 * $\size {\map {y^*} {x - \tilde a} - \map y {x - \tilde a} }_1 \to 0 \implies \epsilon \to 0$

Then $\tilde M$ is conjugate to $M$.