Definition:Conjugate Point (Calculus of Variations)/Definition 3

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

Let:


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


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

Let both $y = \map y x$ and $y = \map {\tilde y} x$ pass through the point $M$.

Let


 * $\displaystyle \lim_{\size {\map y x - \map {\tilde y} x}_1 \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$

In other words, let $\tilde M$ be the limit points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\size {\map y x - \map {\tilde y} x}_1 \to 0$.

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