Equivalence of Definitions of Conjugate Point

Theorem

Definitions of Conjugate Point are equivalent

Proof

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

$\Box$

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.27$: Jacobi's Necessary Condition. More on Conjugate Points