Equivalence of Definitions of Conjugate Point

Defintion 2 implies Definition 3
Let the extremal $y = \map y x$ satisfy:


 * $\map y a = A$

Let the extremal $y^*_\alpha$ pass through $M = \tuple {a, A}$ and satisfy:


 * $\map { {y^*}'_\alpha} a - \map {y'} a = \alpha$

Then the following is a valid expression for $y^*_\alpha$:


 * $\map {y^*_\alpha} x = \map y x + \map h x \alpha + \epsilon$

where $\epsilon = k \alpha$ with:


 * $\alpha \to 0 \implies k \to 0$

and:


 * $\map h a = 0$
 * $\map {h'} a = 1$

where $\map h x$ is an appropriate solution to Jacobi's equation.

Let:


 * $\map h {\tilde a} = 0$
 * $\beta = \sqrt {\dfrac \epsilon \alpha}$

Then:


 * $\map h x \ne 0 \quad \forall x \in \openint a b \implies \map {h'} a \ne 0$

By Taylor's Theorem, the expression:


 * $\map {y_\alpha} x - \map y x = \map h x \alpha + \epsilon$

takes values with different signs at the points $\tilde a - \beta$ and $\tilde a + \beta$.

Since:


 * $\alpha \to 0 \implies \beta \to 0$

the limit of points of intersection of $y = \map {y_\alpha^*} x$ and $y = \map y x$, as $\alpha \to 0$ is $\map {\tilde M} {\tilde a, \map y {\tilde a} }$