Simple Variable End Point Problem

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Theorem

Let $y$ and $F$ be mappings.

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Suppose the endpoints of $y$ lie on two given vertical lines $x = a$ and $x = b$.

Suppose $J$ is a functional of the form

$(1): \quad J \sqbrk y = \ds \int_a^b \map F {x, y, y'} \rd x$

and has an extremum for a certain function $\hat y$.

Then $y$ satisfies the system of equations

$\begin {cases}

F_y - \dfrac \d {\d x} F_{y'} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = a} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = b} = 0 \end {cases}$

Proof

From Condition for Differentiable Functional to have Extremum we have

$\bigvalueat {\delta J \sqbrk {y; h} } {y \mathop = \hat y} = 0$

The variation exists if $J$ is a differentiable functional.

We will start from the increment of a functional:

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$\ds \Delta J \sqbrk {y; h}$

$=$

$\ds J \sqbrk {y + h} - J \sqbrk y$

definition

$\ds $

$=$

$\ds \int_a^b \map F {x, y + h, y' + h'} \rd x - \int_a^b \map F {x, y, y'} \rd x$

$(1)$

$\ds $

$=$

$\ds \int_a^b \paren {\map F {x, y + h, y' + h'} - \map F {x, y, y'} } \rd x$

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Using multivariate Taylor's theorem, one can expand $\map F {x, y + h, y' + h'}$ with respect to $h$ and $h'$:

$\map F {x, y + h, y' + h'} = \bigvalueat {\map F {x, y + h, y' + h'} } {h \mathop = 0, \, h' \mathop = 0} + \valueat {\dfrac {\partial \map F {x, y + h, y' + h'} } {\partial y} } {h \mathop = 0, \, h' \mathop = 0} h + \valueat {\dfrac {\partial {\map F {x, y + h, y' + h'} } } {\partial y'} } {h \mathop = 0, \, h' \mathop = 0} h' + \map \OO {h^2, h h', h'^2}$

Substitute this back into the integral.

Note that the first term in the expansion and the negative one in the integral will cancel out:

$\ds \Delta J \paren {y; h} = \int_a^b \paren {\map F {x, y, y'}_y h + \map F {x, y, y'}_{y'} h' + \map \OO {h^2, h h', h'^2} } \rd x$

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Terms in $\map \OO {h^2, h'^2}$ represent terms of order higher than 1 with respect to $h$ and $h'$.

Now we expand:

$\ds \int_a^b\map \OO {h^2, h h', h'^2} \rd x$

Every term in this expansion will be of the form:

$\ds \int_a^b \map A {m, n} \frac {\partial^{m + n} \map F {x, y, y'} } {\partial y^m \partial {y'}^n} h^m h'^n \rd x$

where $m, n \in \N$ and $m + n \ge 2$

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By definition, the integral not counting in $\map \OO {h^2, h h', h'^2}$ is a variation of functional.

$\ds \delta J \sqbrk {y; h} = \int_a^b \paren {F_y h + F_{y'} h'} \rd x$

Now, integrate by parts and note that $\map h x$ does not necessarily vanish at the endpoints:

$\ds \delta J \sqbrk {y; h}$

$=$

$\ds \int_a^b \paren {F_y - \frac \d {\d x} F_{y'} } \map h x \rd x + \bigintlimits {F_{y'} \map h x} {x \mathop = a} {x \mathop = b}$

$\ds $

$=$

$\ds \int_a^b \paren {F_y - \frac \d {\d x} F_{y'} } \map h x \rd x + \bigvalueat {F_{y'} } {x \mathop = b} \map h b - \bigvalueat {F_{y'} } {x \mathop = a} \map h a$

Then, for arbitrary $\map h x$, $J$ has an extremum if:

$ \begin {cases}

F_y - \dfrac \d {\d x} F_{y'} = 0 \\ \bigvalueat {F_{y'} } {x \mathop = a} = 0\\ \bigvalueat {F_{y'} } {x \mathop = b} = 0 \end {cases}$

$\blacksquare$

Sources

1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 1.6$: A Simple Variable End Point Problem