Definition:Conjugate Point

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Definition

Geometry

Let $\KK$ be a conic section.

Let $P$ and $Q$ be points in the plane of $\KK$.

Let:

$P$ lie on the polar of $Q$
$Q$ lie on the polar of $P$.


$P$ and $Q$ are known as conjugate points with respect to $\KK$.


Calculus of Variations

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.