Interpreting Volterra Series CorrectlyWhat is $int frac{delta F}{delta u} frac{delta G}{delta v} , dx ;...
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Interpreting Volterra Series Correctly
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$begingroup$
I was reading this and found on page 6 was a description of a generalization of Taylor series to linear functionals. Reproduced below as
$$ F[phi + lambda] = sum_{n=0}^{infty} left[frac{1}{n!} intint...int frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) } dx^0 dx^1 ... dx^{n-1}
right] $$
I don't understand exactly what the term $$ frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) }$$
Means.
And let me be extremely concrete. Consider an operator $F: phi rightarrow phi + x^2phi'$, quite literally what this means is that $F: (mathbb{R} rightarrow mathbb{R}) rightarrow (mathbb{R} rightarrow mathbb{R}) $
And $F:= phi(x) rightarrow phi(x) + x^2 frac{d}{dx}left[ phi(x)right] $ Can be viewed as an explicit "form".
So when we compute $$frac{delta F}{delta phi(x)}$$
Everything makes sense (since the argument of the $phi$ we are differentiating w.r.t matches the argument of the $phi$ in the original function and the corresponding $frac{d}{dx}$ is compatible with the $x$ arguments in both cases.) And we use our typical euler lagrange equations.
$$ frac{delta F}{delta phi(x)} = frac{partial F}{partial phi(x)} - frac{d}{dx} left[ frac{partial F}{partial left( frac{d}{dx}[phi(x)] right) } ...right] = 1+2x $$
If you make an expression like
$$frac{delta F}{delta phi(x^0)}$$
Now it's unclear what you mean, but we can guess. Let us substitute every instance of the string "$x$" with "$x^0$" in $F$ and then this statement can be still well defined. And we conclude that it is equal to
$$ 1 + 2x^0 $$
Now consider:
$$frac{delta^2 F}{delta phi(x^0) delta phi(x^1) }$$
This is now getting us into bad territory. Not only do we not have a compatible variable $x$ in either of arguments but we cannot implicitly "guess" what string substitution needs to be made here since any global level string subtitution will render the other derivative to 0. So I am completely lost as to what this could mean.
physics euler-lagrange-equation functional-calculus
$endgroup$
add a comment |
$begingroup$
I was reading this and found on page 6 was a description of a generalization of Taylor series to linear functionals. Reproduced below as
$$ F[phi + lambda] = sum_{n=0}^{infty} left[frac{1}{n!} intint...int frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) } dx^0 dx^1 ... dx^{n-1}
right] $$
I don't understand exactly what the term $$ frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) }$$
Means.
And let me be extremely concrete. Consider an operator $F: phi rightarrow phi + x^2phi'$, quite literally what this means is that $F: (mathbb{R} rightarrow mathbb{R}) rightarrow (mathbb{R} rightarrow mathbb{R}) $
And $F:= phi(x) rightarrow phi(x) + x^2 frac{d}{dx}left[ phi(x)right] $ Can be viewed as an explicit "form".
So when we compute $$frac{delta F}{delta phi(x)}$$
Everything makes sense (since the argument of the $phi$ we are differentiating w.r.t matches the argument of the $phi$ in the original function and the corresponding $frac{d}{dx}$ is compatible with the $x$ arguments in both cases.) And we use our typical euler lagrange equations.
$$ frac{delta F}{delta phi(x)} = frac{partial F}{partial phi(x)} - frac{d}{dx} left[ frac{partial F}{partial left( frac{d}{dx}[phi(x)] right) } ...right] = 1+2x $$
If you make an expression like
$$frac{delta F}{delta phi(x^0)}$$
Now it's unclear what you mean, but we can guess. Let us substitute every instance of the string "$x$" with "$x^0$" in $F$ and then this statement can be still well defined. And we conclude that it is equal to
$$ 1 + 2x^0 $$
Now consider:
$$frac{delta^2 F}{delta phi(x^0) delta phi(x^1) }$$
This is now getting us into bad territory. Not only do we not have a compatible variable $x$ in either of arguments but we cannot implicitly "guess" what string substitution needs to be made here since any global level string subtitution will render the other derivative to 0. So I am completely lost as to what this could mean.
physics euler-lagrange-equation functional-calculus
$endgroup$
add a comment |
$begingroup$
I was reading this and found on page 6 was a description of a generalization of Taylor series to linear functionals. Reproduced below as
$$ F[phi + lambda] = sum_{n=0}^{infty} left[frac{1}{n!} intint...int frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) } dx^0 dx^1 ... dx^{n-1}
right] $$
I don't understand exactly what the term $$ frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) }$$
Means.
And let me be extremely concrete. Consider an operator $F: phi rightarrow phi + x^2phi'$, quite literally what this means is that $F: (mathbb{R} rightarrow mathbb{R}) rightarrow (mathbb{R} rightarrow mathbb{R}) $
And $F:= phi(x) rightarrow phi(x) + x^2 frac{d}{dx}left[ phi(x)right] $ Can be viewed as an explicit "form".
So when we compute $$frac{delta F}{delta phi(x)}$$
Everything makes sense (since the argument of the $phi$ we are differentiating w.r.t matches the argument of the $phi$ in the original function and the corresponding $frac{d}{dx}$ is compatible with the $x$ arguments in both cases.) And we use our typical euler lagrange equations.
$$ frac{delta F}{delta phi(x)} = frac{partial F}{partial phi(x)} - frac{d}{dx} left[ frac{partial F}{partial left( frac{d}{dx}[phi(x)] right) } ...right] = 1+2x $$
If you make an expression like
$$frac{delta F}{delta phi(x^0)}$$
Now it's unclear what you mean, but we can guess. Let us substitute every instance of the string "$x$" with "$x^0$" in $F$ and then this statement can be still well defined. And we conclude that it is equal to
$$ 1 + 2x^0 $$
Now consider:
$$frac{delta^2 F}{delta phi(x^0) delta phi(x^1) }$$
This is now getting us into bad territory. Not only do we not have a compatible variable $x$ in either of arguments but we cannot implicitly "guess" what string substitution needs to be made here since any global level string subtitution will render the other derivative to 0. So I am completely lost as to what this could mean.
physics euler-lagrange-equation functional-calculus
$endgroup$
I was reading this and found on page 6 was a description of a generalization of Taylor series to linear functionals. Reproduced below as
$$ F[phi + lambda] = sum_{n=0}^{infty} left[frac{1}{n!} intint...int frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) } dx^0 dx^1 ... dx^{n-1}
right] $$
I don't understand exactly what the term $$ frac{delta ^n F[phi] }{delta phi(x^0) delta phi(x^1) ... delta phi(x^{n-1}) }$$
Means.
And let me be extremely concrete. Consider an operator $F: phi rightarrow phi + x^2phi'$, quite literally what this means is that $F: (mathbb{R} rightarrow mathbb{R}) rightarrow (mathbb{R} rightarrow mathbb{R}) $
And $F:= phi(x) rightarrow phi(x) + x^2 frac{d}{dx}left[ phi(x)right] $ Can be viewed as an explicit "form".
So when we compute $$frac{delta F}{delta phi(x)}$$
Everything makes sense (since the argument of the $phi$ we are differentiating w.r.t matches the argument of the $phi$ in the original function and the corresponding $frac{d}{dx}$ is compatible with the $x$ arguments in both cases.) And we use our typical euler lagrange equations.
$$ frac{delta F}{delta phi(x)} = frac{partial F}{partial phi(x)} - frac{d}{dx} left[ frac{partial F}{partial left( frac{d}{dx}[phi(x)] right) } ...right] = 1+2x $$
If you make an expression like
$$frac{delta F}{delta phi(x^0)}$$
Now it's unclear what you mean, but we can guess. Let us substitute every instance of the string "$x$" with "$x^0$" in $F$ and then this statement can be still well defined. And we conclude that it is equal to
$$ 1 + 2x^0 $$
Now consider:
$$frac{delta^2 F}{delta phi(x^0) delta phi(x^1) }$$
This is now getting us into bad territory. Not only do we not have a compatible variable $x$ in either of arguments but we cannot implicitly "guess" what string substitution needs to be made here since any global level string subtitution will render the other derivative to 0. So I am completely lost as to what this could mean.
physics euler-lagrange-equation functional-calculus
physics euler-lagrange-equation functional-calculus
edited Feb 2 at 23:22
frogeyedpeas
asked Feb 2 at 23:05
frogeyedpeasfrogeyedpeas
7,66872054
7,66872054
add a comment |
add a comment |
2 Answers
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$begingroup$
I'll run through your example and hopefully that makes it clear.
$$phimapsto F[phi]= phi+x^2phi'$$
Vary this
$$phi+deltaphimapsto F[phi]+deltaphi+x^2deltaphi'$$
So we have formally
$$frac{F[phi+deltaphi](x)-F[phi](x)}{deltaphi(y)}=frac{deltaphi(x)}{deltaphi(y)}+x^2frac{deltaphi'(x)}{deltaphi(y)}$$
Then we take as our definition that (and more generally would have ignored $mathcal O(deltaphi)$ terms if we have things like $F[phi]=phi^2$)
$$
frac{deltaphi(x)}{deltaphi(y)}:=delta(x-y),
$$
the Dirac delta function. Similarly,
$$
frac{deltaphi'(x)}{deltaphi(y)}:=delta'(x-y).
$$
One can make this rigorous with compactly supported test functions and the like.
$endgroup$
add a comment |
$begingroup$
FWIW, in OP's example OP is considering the functional
$$ F[phi]~:=~ phi(x_0) + x_0^2phi^{prime}(x_0), $$
which depends on the parameter $x^0inmathbb{R}$.
Then the functional/variational derivative is
$$ frac{delta F[phi]}{delta phi(x_1)}~=~delta(x_0!-!x_1) + x_0^2delta^{prime}(x_0!-!x_1), $$
which doesn't depend on $phi$. Therefore the higher functional derivatives vanish.
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
oldest
votes
active
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votes
active
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votes
$begingroup$
I'll run through your example and hopefully that makes it clear.
$$phimapsto F[phi]= phi+x^2phi'$$
Vary this
$$phi+deltaphimapsto F[phi]+deltaphi+x^2deltaphi'$$
So we have formally
$$frac{F[phi+deltaphi](x)-F[phi](x)}{deltaphi(y)}=frac{deltaphi(x)}{deltaphi(y)}+x^2frac{deltaphi'(x)}{deltaphi(y)}$$
Then we take as our definition that (and more generally would have ignored $mathcal O(deltaphi)$ terms if we have things like $F[phi]=phi^2$)
$$
frac{deltaphi(x)}{deltaphi(y)}:=delta(x-y),
$$
the Dirac delta function. Similarly,
$$
frac{deltaphi'(x)}{deltaphi(y)}:=delta'(x-y).
$$
One can make this rigorous with compactly supported test functions and the like.
$endgroup$
add a comment |
$begingroup$
I'll run through your example and hopefully that makes it clear.
$$phimapsto F[phi]= phi+x^2phi'$$
Vary this
$$phi+deltaphimapsto F[phi]+deltaphi+x^2deltaphi'$$
So we have formally
$$frac{F[phi+deltaphi](x)-F[phi](x)}{deltaphi(y)}=frac{deltaphi(x)}{deltaphi(y)}+x^2frac{deltaphi'(x)}{deltaphi(y)}$$
Then we take as our definition that (and more generally would have ignored $mathcal O(deltaphi)$ terms if we have things like $F[phi]=phi^2$)
$$
frac{deltaphi(x)}{deltaphi(y)}:=delta(x-y),
$$
the Dirac delta function. Similarly,
$$
frac{deltaphi'(x)}{deltaphi(y)}:=delta'(x-y).
$$
One can make this rigorous with compactly supported test functions and the like.
$endgroup$
add a comment |
$begingroup$
I'll run through your example and hopefully that makes it clear.
$$phimapsto F[phi]= phi+x^2phi'$$
Vary this
$$phi+deltaphimapsto F[phi]+deltaphi+x^2deltaphi'$$
So we have formally
$$frac{F[phi+deltaphi](x)-F[phi](x)}{deltaphi(y)}=frac{deltaphi(x)}{deltaphi(y)}+x^2frac{deltaphi'(x)}{deltaphi(y)}$$
Then we take as our definition that (and more generally would have ignored $mathcal O(deltaphi)$ terms if we have things like $F[phi]=phi^2$)
$$
frac{deltaphi(x)}{deltaphi(y)}:=delta(x-y),
$$
the Dirac delta function. Similarly,
$$
frac{deltaphi'(x)}{deltaphi(y)}:=delta'(x-y).
$$
One can make this rigorous with compactly supported test functions and the like.
$endgroup$
I'll run through your example and hopefully that makes it clear.
$$phimapsto F[phi]= phi+x^2phi'$$
Vary this
$$phi+deltaphimapsto F[phi]+deltaphi+x^2deltaphi'$$
So we have formally
$$frac{F[phi+deltaphi](x)-F[phi](x)}{deltaphi(y)}=frac{deltaphi(x)}{deltaphi(y)}+x^2frac{deltaphi'(x)}{deltaphi(y)}$$
Then we take as our definition that (and more generally would have ignored $mathcal O(deltaphi)$ terms if we have things like $F[phi]=phi^2$)
$$
frac{deltaphi(x)}{deltaphi(y)}:=delta(x-y),
$$
the Dirac delta function. Similarly,
$$
frac{deltaphi'(x)}{deltaphi(y)}:=delta'(x-y).
$$
One can make this rigorous with compactly supported test functions and the like.
edited Mar 18 at 12:02
answered Feb 3 at 0:06
Alec B-GAlec B-G
52019
52019
add a comment |
add a comment |
$begingroup$
FWIW, in OP's example OP is considering the functional
$$ F[phi]~:=~ phi(x_0) + x_0^2phi^{prime}(x_0), $$
which depends on the parameter $x^0inmathbb{R}$.
Then the functional/variational derivative is
$$ frac{delta F[phi]}{delta phi(x_1)}~=~delta(x_0!-!x_1) + x_0^2delta^{prime}(x_0!-!x_1), $$
which doesn't depend on $phi$. Therefore the higher functional derivatives vanish.
$endgroup$
add a comment |
$begingroup$
FWIW, in OP's example OP is considering the functional
$$ F[phi]~:=~ phi(x_0) + x_0^2phi^{prime}(x_0), $$
which depends on the parameter $x^0inmathbb{R}$.
Then the functional/variational derivative is
$$ frac{delta F[phi]}{delta phi(x_1)}~=~delta(x_0!-!x_1) + x_0^2delta^{prime}(x_0!-!x_1), $$
which doesn't depend on $phi$. Therefore the higher functional derivatives vanish.
$endgroup$
add a comment |
$begingroup$
FWIW, in OP's example OP is considering the functional
$$ F[phi]~:=~ phi(x_0) + x_0^2phi^{prime}(x_0), $$
which depends on the parameter $x^0inmathbb{R}$.
Then the functional/variational derivative is
$$ frac{delta F[phi]}{delta phi(x_1)}~=~delta(x_0!-!x_1) + x_0^2delta^{prime}(x_0!-!x_1), $$
which doesn't depend on $phi$. Therefore the higher functional derivatives vanish.
$endgroup$
FWIW, in OP's example OP is considering the functional
$$ F[phi]~:=~ phi(x_0) + x_0^2phi^{prime}(x_0), $$
which depends on the parameter $x^0inmathbb{R}$.
Then the functional/variational derivative is
$$ frac{delta F[phi]}{delta phi(x_1)}~=~delta(x_0!-!x_1) + x_0^2delta^{prime}(x_0!-!x_1), $$
which doesn't depend on $phi$. Therefore the higher functional derivatives vanish.
answered Feb 3 at 12:05
QmechanicQmechanic
5,17711858
5,17711858
add a comment |
add a comment |
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