Help me understand convolutions and Green's functions

[jack476]

I'm working through the problems in the first chapter of Jackson and I'm still grappling with the interpretation of Green's functions.

I understand that if I have the Poisson equation $\nabla^2\phi(x) = \frac{-\rho (x)}{\epsilon_0}$ and the Green's function $G(x, x^\prime)$ then in general $\phi(x) = \frac{\rho (x)}{\epsilon_0}\ast G(x, x^\prime)$.

What's bothering me is that, if my understanding is correct, this would give $\phi(x) = \frac{1}{\epsilon_0}\int \rho(x^\prime)G(x,x^\prime)dV^\prime$. But this is missing a factor of $\frac{1}{4\pi}$, for instance, in the case of a point charge where Jackson says that we have $G = \frac{1}{|x-x^\prime|}$ then we have $\phi(x) = \frac{1}{\epsilon_0}\int \frac{\rho(x^\prime)}{|x-x^{\prime}|}dV^\prime$, which is not correct because it's missing that factor.

The only way I can see it is that, properly speaking, for a linear differential operator $L$ a Green's function is a solution to $LG(x, x^\prime) = \delta (x-x^\prime)$, but Jackson has defined Green's functions to be solutions to the differential equation $\nabla^2 G(x,x^\prime) = -4\pi \delta(x-x^\prime)$, so that $G$ is not a Green's function for the Laplacian but rather for the operator $\frac{-\nabla^2}{4\pi}$. If we represent Poisson's equation in terms of this operator by multiplying each side by $\frac{-1}{4\pi}$ then we get $\frac{-\nabla^2}{4\pi}\phi(x) = \frac{\rho(x)}{4\pi \epsilon_0}$ and now we can use the convolution to get the correct formula: $\phi(x) = \frac{\rho (x)}{4\pi \epsilon_0}\ast G(x, x^\prime) =\frac{1}{4\pi \epsilon_0}\int \rho(x^\prime)G(x, x^\prime)dV^\prime$, but that just seems kind of inelegant.

Can someone please help me understand all this? That missing $\frac{1}{4\pi}$ has been ruining my entire day.

 

[Orodruin]

Regarding signs, you might encounter different conventions regarding the Green's function for Poisson's equation. The most common definition for a linear differential operator $L$ is that the Green's function solves $L G(x,x') = \delta(x-x')$ and for Poisson's equation you can choose to find the Green's function of either $\nabla^2$ or $-\nabla^2$. As you have already discovered, the only difference is in how you build the final solution from your Green's function. Assuming the former definition (using $\nabla^2$, not $-\nabla^2$), the Green's function of Poisson's equation in three dimensions is
$$
G(\vec x, \vec x') = - \frac{1}{4\pi} \frac{1}{|\vec x - \vec x'|}.
$$
The general result for $N$ dimensions is also pretty straight-forward to derive. You should find this in any textbook covering Green's functions in more than one dimension as it is a rather standard case.

 

[SchroedingersLion]

Not on topic, but a good quote I remember:
"Learning electrodynamics with Jackson is like learning to swim in a bucket of cement"

 

[jack476]

Having slept on it, I think that I now understand it.

Suppose that $\phi({x^\prime})$ is the potential at $x^\prime$. From the properties of the delta function, the potential at $x$ due to the potential at $x^\prime$ is equal to $\int \phi(x^\prime)\delta(x-x^\prime)dV^\prime$, which is a convolution with an impulse. From Jackson's definition, this is equal to $\frac{-1}{4\pi}\int \phi(x^\prime)\nabla^2G(x,x^\prime)dV^\prime$. By Green's Theorem, we have $\int \left( \frac{-\phi(x^\prime)}{4\pi}\nabla^2G(x,x^\prime)+\frac{G(x, x^\prime)}{4\pi}\nabla^2\phi(x^\prime)\right)dV^\prime = \frac{1}{4\pi}\oint\left( \phi(x^\prime)\frac{\partial G(x, x^\prime)}{\partial n^\prime}-G(x, x^\prime)\frac{\partial \phi(x^\prime)}{\partial n^\prime}\right)da^\prime$ and by substituting in Poisson's equation and rearranging we have $\int \left( \frac{-\phi(x^\prime)}{4\pi}\nabla^2G(x,x^\prime)\right)dV^\prime = \frac{1}{4\pi \epsilon_0}\int G(x, x^\prime)\rho(x^\prime)dV^\prime + \frac{1}{4\pi}\oint\left( \phi(x^\prime)\frac{\partial G(x, x^\prime)}{\partial n^\prime}-G(x, x^\prime)\frac{\partial \phi(x^\prime)}{\partial n^\prime}\right)da^\prime$ and the left hand side of this equation is just $\phi(x)$, and this is the formula that Jackson provides on page 39. So I guess in that sense, the right hand side is the convolution operator, and $\phi$ is not explicitly a convolution with $G$ but rather a convolution with an impulse in which $G$ appears implicitly.

Regarding signs, you might encounter different conventions regarding the Green's function for Poisson's equation. The most common definition for a linear differential operator $L$ is that the Green's function solves $L G(x,x') = \delta(x-x')$ and for Poisson's equation you can choose to find the Green's function of either $\nabla^2$ or $-\nabla^2$. As you have already discovered, the only difference is in how you build the final solution from your Green's function. Assuming the former definition (using $\nabla^2$, not $-\nabla^2$), the Green's function of Poisson's equation in three dimensions is
$$
G(\vec x, \vec x') = - \frac{1}{4\pi} \frac{1}{|\vec x - \vec x'|}.
$$
The general result for $N$ dimensions is also pretty straight-forward to derive. You should find this in any textbook covering Green's functions in more than one dimension as it is a rather standard case.

I want to know more about this but I'm not sure if I have the time. Do you have any recommendations that can be covered in a reasonably short time and also aren't very expensive?

Not on topic, but a good quote I remember:
"Learning electrodynamics with Jackson is like learning to swim in a bucket of cement"

I'm actually not finding it to be as bad as everyone says. I'm pretty comfortable with PDEs and complex variables because I took a lot of extra math in undergrad, and also I did very well in my undergrad E&M courses, and I like that the problems are tough and require a lot of thinking. My only complaint is that, so far, it seems to want to be an applied math textbook more than a physics book. That's why I was so hung up on the Green's functions, I was trying to understand what the physical motivation is for them.

 

[Orodruin]

I want to know more about this but I'm not sure if I have the time. Do you have any recommendations that can be covered in a reasonably short time and also aren't very expensive?

I would recommend chapter 7 of my book (should be around 60 pages, but covers things starting from Green’s functions in one dimension as well as boundary conditions), but I doubt that would qualify for your ”aren’t very expensive” criterion if that is all you would want it for. (I am also obviously biased...) I don’t really know of any other short and concise treatment of Green’s functions at the same level. I did not really like how Green’s functions were treated in other mathematical methods text (partially the reason for writing my own).

For the inexpensive route, I would suggest starting out using the Wiki pages and asking questions here should the need arise.

 

[jack476]

I would recommend chapter 7 of my book (should be around 60 pages, but covers things starting from Green’s functions in one dimension as well as boundary conditions), but I doubt that would qualify for your ”aren’t very expensive” criterion if that is all you would want it for. (I am also obviously biased...) I don’t really know of any other short and concise treatment of Green’s functions at the same level. I did not really like how Green’s functions were treated in other mathematical methods text (partially the reason for writing my own).

For the inexpensive route, I would suggest starting out using the Wiki pages and asking questions here should the need arise.

Thank you for the recommendation, I found a copy in my school's interlibrary loan network so I will look into that. Sorry to deny you a sale though, but money's tight right now.

 

[Orodruin]

Thank you for the recommendation, I found a copy in my school's interlibrary loan network so I will look into that. Sorry to deny you a sale though, but money's tight right now.

Honestly, the money I would get for a sale is not life-changing.
I am happy to hear you found it in your school’s library because it means some libraries have it. My own university has several e-library copies that I usually try to point my own students to.

 

[vanhees71]

Not on topic, but a good quote I remember:
"Learning electrodynamics with Jackson is like learning to swim in a bucket of cement"

To the contrary, if you understand a topic in Jackson's book you have a rather complete knowledge about it. Jackson's book is tough, but in the long term it saves a lot of time wasted with books not as precise as Jackson. To my taste Jackson is a bit too traditional in its approach though, because it takes several chapters until he comes to the relativistic formulation, which is the most natural one to begin with. My favorite at this level of textbooks rather is Landau&Lifshitz vol. II. As Jackson it's not for beginners. Here I'd recommend Sommerfeld vol. III (of course with the caveat that he uses the $\mathrm{i} c t$ convention for the Minkowski fundamental form) or the Feynman Lectures vol. II.

 

[jack476]

To the contrary, if you understand a topic in Jackson's book you have a rather complete knowledge about it. Jackson's book is tough, but in the long term it saves a lot of time wasted with books not as precise as Jackson. To my taste Jackson is a bit too traditional in its approach though, because it takes several chapters until he comes to the relativistic formulation, which is the most natural one to begin with. My favorite at this level of textbooks rather is Landau&Lifshitz vol. II. As Jackson it's not for beginners. Here I'd recommend Sommerfeld vol. III (of course with the caveat that he uses the $\mathrm{i} c t$ convention for the Minkowski fundamental form) or the Feynman Lectures vol. II.

On this note, a couple months before I started Jackson I worked through a book by O'Hanian which was also called Classical Electrodynamics, and although it was at the level of Wangsness or Griffiths it used the covariant formulation to development magnetism immediately after finishing the first few chapters on electrostatics. The presentation of the idea that magnetism occurs due to the Lorentz transformations of the electromagnetic field tensor between frames was incredibly elegant and kind of a mindblower for me.

 

[SchroedingersLion]

To the contrary, if you understand a topic in Jackson's book you have a rather complete knowledge about it. Jackson's book is tough, but in the long term it saves a lot of time wasted with books not as precise as Jackson. To my taste Jackson is a bit too traditional in its approach though, because it takes several chapters until he comes to the relativistic formulation, which is the most natural one to begin with. My favorite at this level of textbooks rather is Landau&Lifshitz vol. II. As Jackson it's not for beginners. Here I'd recommend Sommerfeld vol. III (of course with the caveat that he uses the $\mathrm{i} c t$ convention for the Minkowski fundamental form) or the Feynman Lectures vol. II.

I am not saying Jackson sucks. It only sucks for complete beginners in their first or second year. Our professor also described it as "the bible of electrodynamics". But he also said it would be very hard to swallow for newbies. And from a didactic perspective, the high and rigorous level of this book might be too discouraging to beginners.

 

[vanhees71]

Of course, Jackson is not written for first- or second-year students...