Definition:Conjugate Point (Calculus of Variations)

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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.


Definition 2

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 $M$.


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

$\ds \lim_{\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$

In other words, let $\tilde M$ be the limit points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0$.



Then $\tilde M$ is conjugate to $M$.


Dependent on $N$ Functions

Let $K$ be a functional such that:

$\ds K \sqbrk h = \int_a^b \paren {\mathbf h'\mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$

Consider Euler's equation related to the functional $K$:

$-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = 0$

where $\mathbf P$ and $\mathbf Q$ are symmetric matrices.

Let the general solution to this equation be:

$\set {\mathbf h^{\paren i} = \paren {\sequence {h_{ij} } }: i,j \in \N_{\le N} }$

Let:

$\exists j: \forall k \ne j: \paren {\map {\mathbf h^{\paren j} } a = 0} \land \paren {\map {h_{j j}'} a = 1, h'_{j k} = 0}$

Let the determinant, built from $h_{ij}$, be such that:

$\size {h_{i j} } \paren {\tilde a} = 0$

Here $i$ denotes rows, and $j$ denotes columns.


Then $\tilde a$ is said to be conjugate to point $a$ with respect to the functional $K$.


With Respect to Original Functional

Let:

$\ds \int_a^b \map F {x, y, y'}$

be the original functional.

Let $\tilde a$ be conjugate to $a$.

Let:

$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$

be the second variation of $\ds \int_a^b \map F {x, y, y'}$.


Then $\tilde a$ is conjugate to $a$ with respect to to the original functional $\ds \int_a^b \map F {x, y, y'}$.


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 $\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$.


Sources