Definition:Conjugate Point/Definition 3
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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
- $\displaystyle \lim_{\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$
In other words, let $\tilde M$ be the limit points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0$.
the notation $\norm {\map y x - \map {\tilde y} x}_{1, \infty}$ with particular regard to the subscript
This is most likely the standard norm of Sobolev space, also know as np-norm. Original source never calls it by word; only by special symbols
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Then $\tilde M$ is conjugate to $M$.
Sources
As such, the following source works, along with any process flow, will need to be reviewed.
Specifically: with respect to definitions 1, 2 and 3
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- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.26$: Analysis of the Quadratic Functional $ \int_a^b \paren {P h'^2 + Q h^2} \rd x$