What does this proof mean? (variation of high-order derivative)

[thaiqi]

TL;DR

A question in calculus of variation.

I read in one book proving one nature of variation(variation of high-order derivative).
It writes that "$\delta(F^{(n)}) = F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)} = (\delta F)^{(n)}$".
But I don't understand where this $F_0$ comes out from.

 

[jedishrfu]

Isn't $F_0$ the starting point of the variation?

 

[thaiqi]

Isn't $F_0$ the starting point of the variation?

Sorry I don't catch you well. Do you mean it is the "starting point"? What would it then mean?

 

[jedishrfu]

When one integrates a function, you might go from x=a to x=b

$\int_a^b f(x)dx = F(x)|_a^b = F(b) - F(a)$

and so it looks like x=a is the $x_0$ and $F(a) = F_0$

 

[thaiqi]

Summary:: A question in calculus of variation.

I read in one book proving one nature of variation(variation of high-order derivative).
It writes that "$\delta(F^{(n)}) = F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)} = (\delta F)^{(n)}$".
But I don't understand where this $F_0$ comes out from.

The right part is understandable. But the left part: $F^{(n)} - F_0^{(n)} = \delta(F^{(n)})$, does it hold?

 

[jedishrfu]

Isn't that just the definition of a variation?

The key part is the change to $F^{(n)} - F_0^{(n)} = (F - F_0)^{(n)}$

Jedi calling @fresh_42 --> Come in @fresh_42

 

[fresh_42]

Analysis is a local theory. All statements are about a certain local point. In this case we have $F_0^{(n)}$ as point of interest. $F^{(n)}$ is $F_0^{(n)}$ plus a little bit off. This little bit is $\delta(F^{(n)})$.

 

[thaiqi]

Another place in this book it writes:
"
$\delta(\int_{x_0}^{x_1}Fdx) = {\partial \over \partial y}(\int_{x_0}^{x_1}Fdx)\delta y + {\partial \over \partial y^\prime}(\int_{x_0}^{x_1}Fdx)\delta y^\prime$
$= (\int_{x_0}^{x_1}F_y dx)\delta y + (\int_{x_0}^{x_1}F_{y^\prime}dx)\delta y^\prime$
$= \int_{x_0}^{x_1}(F_y \delta y + F_{y^\prime}\delta y^\prime) dx$
$= \int_{x_0}^{x_1} \delta F dx$
"
My question is: how does the third equal sign hold? Aren't $\delta y$ and $\delta y^\prime$ functions of $x$, how can they be moved into the integral?

 

[thaiqi]

To complement, it writes this in front of the above place:

$\delta F = F_y \delta y + F_y^{\prime} \delta y^{\prime}$

 

[WWGD]

Thaiqi: How is your $\delta$ defined?

 

[thaiqi]

Thaiqi: How is your $\delta$ defined?

$\delta$ is the sign of variation. $\delta y = y(x) - y_0(x) = \epsilon \eta (x)$