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:
$P$ 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.