Timeline for Deriving Klein-Gordon from Hamilton's equations for fields using functional derivatives
Current License: CC BY-SA 4.0
13 events
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| Oct 30, 2020 at 22:38 | history | edited | mike stone | CC BY-SA 4.0 |
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| Oct 30, 2020 at 22:24 | history | edited | mike stone | CC BY-SA 4.0 |
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| Oct 30, 2020 at 22:22 | comment | added | mike stone | I'll add to my answer again. And no, the partial derivatives are not the same as the functional derivatives. I already gave the defintion of the functional derivative in my answer. | |
| Oct 30, 2020 at 22:07 | comment | added | hodop smith | When you write "Comparing with the definition of the functional derivative," and then you use something that looks like the total derivative of a function, how do you get terms like $\frac{\delta H}{\delta\pi}\delta\pi$? Shouldn't those terms be like $\frac{\partial H}{\partial\pi}\delta\pi$. I just looked up the definition of the functional derivative on Wikipedia, and they use the partials just like in the total derivative of a function. I need to use the $\delta$ version if I'm going to do the "read off" you use. Can you cite the definition of the functional derivative, or explain it? | |
| Oct 29, 2020 at 21:45 | history | edited | mike stone | CC BY-SA 4.0 |
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| Oct 29, 2020 at 21:18 | vote | accept | hodop smith | ||
| Oct 30, 2020 at 22:04 | |||||
| Oct 29, 2020 at 21:17 | comment | added | hodop smith | I don't have a book on this subject. Thank you for expanding your answer. The method for comparing the two expressions for $\delta H$ is what I was missing. | |
| Oct 29, 2020 at 21:11 | vote | accept | hodop smith | ||
| Oct 29, 2020 at 21:12 | |||||
| Oct 29, 2020 at 21:08 | comment | added | mike stone | I added text showing how it works. I was assuming you have a book on this subject that you were using. | |
| Oct 29, 2020 at 21:07 | history | edited | mike stone | CC BY-SA 4.0 |
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| Oct 29, 2020 at 21:06 | comment | added | hodop smith | @jacob1729 Yes, I think I can see that my units indicate the integral should go away because, for instance, $\dot\varphi$ does not have units of $d^4x\pi$. However, the answer "read up on functional derivatives" to my question "how do functional derivatives work" is obnoxiously unhelpful. Can you say a little more about what you mean by, "Bring down factors of the Dirac delta?" Thank you. | |
| Oct 29, 2020 at 20:58 | comment | added | jacob1729 | Whilst it's correct that the functional derivatives should have brought down factors of $\delta(x-x')$ which eat up the integrals, I'm not sure this actually answer's OPs question of what to do for $\frac{\delta}{\delta \phi} \int d^4 x \nabla^2 \phi$? | |
| Oct 29, 2020 at 20:50 | history | answered | mike stone | CC BY-SA 4.0 |