Definition:Conjugate Point
Please fix formatting and $\LaTeX$ errors and inconsistencies.
It may also need to be brought up to our standard house style.
If you have done this or know it has been done, please remove {{Tidy}} from the code.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Contents
Definition
Definition 1
Let
- $\displaystyle-\frac \d {\d x} \paren {Ph'}+Qh=0$
with boundary conditions
- $\map h a=0,\quad \map h c=0,\quad a<c\le b$
Suppose
- $\map h x=0\quad\neg\forall x\in\closedint a b$
Suppose
- $\map h a=0,\quad \map h {\tilde a}=0,\quad a\ne\tilde a$
Then the point $\tilde a$ is called conjugate to the point $a$ with respect to solution to the aforementioned differential equation.
Are both a and c conjugate?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Definition 2
Let $y=\map y x$ and $y^*=\map {y^*} x$ be extremal functions.
Let
- $M=\paren{a,\map y a}$
- $\tilde M=\paren{\tilde a,\map y {\tilde a} }$
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 $M$.
Definition 3
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 of 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 $M$.
Also defined as
In the context of Calculus of Variations, functionals are one of the most important concepts.
Therefore, instead of a function, a functional which is minimised by the given function is used as a concept of reference.
Then, if $\tilde a$ is conjugate to $a$ with respect to solution of $\sqbrk {-\dfrac \d {\d x} \paren{Ph'}+Qh=0}$, then it is also conjugate with respect to $\displaystyle\int_a^b \paren{Ph'^2+Qh^2}\rd x$.
Sources
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{Ph'^2+Qh^2}\rd x$
