Definition:Conjugate Point

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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.



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$