Definition:Conjugate Point (Calculus of Variations)/Definition 1

Definition

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.