Definition:Conjugate Point
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Contents
Definition
Definition 1
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.
Are both a and c conjugate?
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Definition 2
Let $y = \map y x$ and $y^* = \map {y^*} x$ be extremal functions.
Let
- $M = \paren {a, \map y a}$
- $\tilde M = \paren {\tilde a, \map y {\tilde a} }$
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 $M$.
Definition 3
Let $y = \map y x$ and $y = \map {\tilde y} x$ be extremal functions.
Let
- $M = \paren {a, \map y a}$
- $\tilde M = \paren {\tilde a, \map y {\tilde a} }$
Let both $y = \map y x$ and $y = \map {\tilde y} x$ pass through the point $M$.
Let
- $\displaystyle \lim_{\size {\map y x - \map {\tilde y} x}_1 \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$
In other words, let $\tilde M$ be the limit of points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\size {\map y x - \map {\tilde y} x}_1 \to 0$
Then $\tilde M$ is conjugate to $M$.
Also defined as
In the context of Calculus of Variations, functionals are one of the most important concepts.
Therefore, instead of a function, a functional which is minimised by the given function is used as a concept of reference.
Then, if $\tilde a$ is conjugate to $a$ with respect to solution of $\paren{-\map {\dfrac \d {\d x} } {P h'} + Q h = 0}$, then it is also conjugate with respect to $\displaystyle \int_a^b \paren {P h'^2 + Q h^2} \rd x$.
Sources
- 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 { Ph'^2 + Q h^2} \rd x$
