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

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## 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.

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## Sources

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- 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$