Green's functions: the genius way to solve DEs

Green's functions is a very powerful and clever technique to solve many differential equations, and since differential equations are the language of lots of physics, including both classical mechanics, electrodynamics, and even quantum field theory, so it is important to know how it works. Of course, this includes some explanation and perhaps a pretty different motivation for Dirac delta function, which is pretty weird, but also not really when you think about it a different way.

Correction: in 19:11, the Green's function lacks a factor of 1/m.

This video simply aims to introduce the Green's functions, what it is supposed to do, how the motivation of it all comes to be, and why it works. If you do need a lot more than introductory knowledge on Green's functions, and you are comfortable in basic differential equation solving, here are some links:

For those who want some answers for the exercise towards the end of Chapter 3, i.e. around 15:47:

Essentially, what I intended was that using that momentum change = integral of force over small period of time, you can obtain the first answer (by a similar definition of delta function in 1D), and I am expecting "point impulse / impulse" on Q2.

For Q3: It is supposed to be that "applied force can be thought of as a 'continuous sum' of point impulses".

For Q4: the Green's function describes the displacement of the oscillator after we apply an impulse. For this reason, Green's function is usually called the "impulse response".

For Q5: Exactly copying the "adding different charge distributions (implies) adding up the electric potential", so in this case, "adding different forces (implies) adding up the displacement"

For Q6: From the formula that x(t) = int G(t, tau)*F(tau) d(tau), we can interpret that the displacement is a continuous sum of the impulse responses.

I stopped saying anything more because (1) the video is already very long, (2) this video assumes only basic knowledge of calculus (it is actually better if you don't know too much of the rigour in real analysis, since this is really hand-wavy - and it has to be! Otherwise this would be a lecture in distribution theory, which I am not quite well-versed in), and (3) this really just aims to provide motivation for Green's functions and doing examples would make this more "textbook-y" than it already is.

Of course, the link to the Wikipedia table of Green's functions:

Note: they don't state the boundary / initial conditions explicitly, and they don't even use x and xi, or t and tau, usually just their difference. Usually it is that the Green's functions vanish when the position is far away from the origin, and for those involving time, 0 before time tau, assuming that tau is greater than 0 (the so-called "advanced" Green's function)

A little bit of remark after viewing the video once again: in some places, it is a little bit quick, so please treat yourself by pausing if necessary: YouTube allows you to do so! In my defense, different people require different time to pause, and also I don't want too much of dead air, so... that is probably also how a lot of other math videos on YouTube are doing right now.

Video chapters:
00:00 Introduction
01:01 Linear differential operators
03:54 Dirac delta "function"
09:56 Principle of Green's functions
15:50 Sadly, DE is not as easy

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[mathemaniac]

Originally I wanted to upload this on 14th July, George Green's birthday, but this took longer than expected, so here we are.

Correction: in 19:11, the Green's function lacks a factor of 1/m.

The omega^2 in the oscillator equation should be replaced by k. Technically I didn't say that omega has to be the angular frequency, but since it normally does, people do point that out, so I'm also pointing it out as well.

[walterufsc]

For those familiar with linear systems analysis, there is a useful analogy: Green's function corresponds to the system's response when the input is an impulse function (Dirac's delta). Thus, to obtain the solution for a different excitation, we use the convolution integral of the impulse response (Green's function) with the input to the system.

[dontsmackdafish]

3Blue1Brown is having a video contest this summer. You should submit this! It's great!

[noahifiv]

I just fell on my head and checked youtube for something that I could watch without having to concentrate to hard. I didn't know about greens function. I managed to follow the video almost to the end :-) I return happily tomorrow when my head is better. thank you for your work.

[benburdick]

I think this is the most approachable video on Green's functions I've ever seen. Thanks for making this! It's going to take a few watches to sink in, but already it's starting to make more sense. Your videos are always super interesting, and extremely helpful!

[sweepsweep]

you know dude even if you do not say it out loud but having been through college maths I can tell everyone that making this video is not easy. For such a crazy high level topic being explained so simply there is easily multiple hours of work put in to generate every minute of video, from scripting, conceptualizing, text and sketches, animation, voicing, music, and ensuring at each stage it is making sense to a newcomer and adding all the required bits in a predigested easy to follow way requires tremendous hard work as well as tremendous effort. He has summarized 6ish hours of maths in 20 minutes and made it accessible to every single person who has even a basic math foundation. Serious hats off dude. You are amazing. Absolutely amazing!

[howkudyou]

This is by far the best video on Green's functions I could find. I'm currently taking Electrodynamics at uni and it helped me finally understand this topic. Thank you!

[washieman]

I really appreciate the fact that you make these videos interesting to those who already know a little bit of math and wish to go a bit deeper in. Thank you.

[henryginn]

I just finished my third year of a maths degree and the intuition that I had gathered for Green's functions was that it was an "infinitesimal amount of solution to the DE" that is integrated over the region. Of course they don't explain anything at all in this aspect so it's nice to see it explained with animations

[PabloAvilaEstevez]

Great stuff man, when I was at the university I found tons of resources for lower division math and physics, but once I started my upper divisions things like these were harder to find, and made in such a comprehensive way at that. Thank you and may you prosper

[lordkelvin]

I've never seen a video giving us such an AMAZING both introduction to green functions and using them. When our teacher for theoretical physics explained us this years ago I only slept in during the lecture. Many, MANY thanks! This video is PERFECT. No more words to say.

[jamespage]

This is the best explanation of Green's functions I've seen, thank you! And the applications are limitless: the propagators in Feynman diagrams are based on Green's functions for example, so if you get this video, you're well set to learn quantum field theory

[markmajkowski]

VERY POWERFUL. When learning Green’s Functions (long forgotten) — after you do enough — you can basically just write down the answer.

[TobyAsE]

Thank you for this flash back to my theoretical electrodynamics lecture. Back when studying physics was kinda fun...

[curiousaboutscience]

Definitely see these everywhere in higher level physics. Great to see the E&M examples!

[soumyadipghosh]

I get to learn a way more (at least geometrically) than from my instructors.
Propagators can be a real nuisance in QM, without understanding what is a green function. Great Explanation! ❤️

[nirajangupta]

I think from the responses you have received it's clear that many of the viewers, if not all, want advance topics to be covered as well. So we hope you will not let us down.BTW, Keep going sir, you are doing a great job 🙂🙂🙂.

[georgemartin]

Mathemaniac, you are one of the best teachers I've ever seen.
Those animations, a visual interpretation of maths could be a key tool for anyone's comprehension capabilities. I may test out if someone from my family without maths background can understand this.

This could be awesome. Wish me luck.

[tehyonglip]

You have done it! You have taught what my lecturers have failed to teach for the whole semester in 23 minutes!

[sachs]

This is, among your videos, the one I could least follow. I don't know physics and, to me, the examples only obfuscated the subject. In general I still love your videos thou, just felt the urge to, once again, modulate your knowledge of our background. So go on! Maybe one day I will come back to this one.