Equivalence of Definitions of Conjugate Point

Theorem
Definitions of Conjugate Point are equivalent

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


 * $\map y a=A$

Let the extremal $y^*_\alpha$ pass through $M=\paren {a,A}$ 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


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

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

Let


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

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 $\tilde M \paren {\tilde a, \map y {\tilde a} }$