# Definition:Conjugate Point

## 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.

### 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$