# Definition:Conjugate Point

## Definition

### Definition 1

Let:

- $-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$

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:

- $M = \tuple {a, \map y a}$

- $\tilde M = \tuple {\tilde a, \map y {\tilde a} }$

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

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

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

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 $\paren {-\map {\dfrac \d {\d x} } {P h'} + Q h = 0}$, then it is also **conjugate** with respect to $\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$.

## 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 {P h'^2 + Q h^2} \rd x$