# Definition:Conjugate Point (Calculus of Variations)/Definition 2

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## Definition

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 $M$.

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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$