Definition:Conjugate Point (Calculus of Variations)/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

$\ds \lim_{\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$

In other words, let $\tilde M$ be the limit points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0$.

This article, or a section of it, needs explaining. In particular: 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 You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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

Sources

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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$