Timeline for Rigorous Euler-Lagrange equations for fields
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 4 at 14:39 | answer | added | Bence Racskó | timeline score: 4 | |
| Feb 5, 2021 at 22:17 | comment | added | Mikhail Skopenkov | A rigorous discussion starts with an accurate statement rather than a proof ("derivation"). One should specify which exactly space of functions $\phi$ is considered and so on. E.g. in the reference recommended by Carlo Beenakker only C^2 smooth fucntions and C^2 smooth variations are considered. But minimizers even of smooth actions often cannot be found among smooth functions. | |
| Feb 2, 2021 at 2:45 | vote | accept | MathMath | ||
| Feb 2, 2021 at 2:40 | comment | added | Pedro Lauridsen Ribeiro | @Buzz the derivation can be made rigorous (see e.g. mathoverflow.net/a/349234/11211, mathoverflow.net/q/273254/11211 and physics.stackexchange.com/a/256496/16767). The problem is just the terminology employed in (some of) the physical literature - one is in fact looking for critical points of a family of action functionals over each compact region of space-time. Once properly defined, these are bona fide smooth functionals on the space of smooth field configurations, which on its turn can be endowed with a proper infinite-dimensional smooth manifold structure. | |
| Feb 1, 2021 at 22:01 | comment | added | Buzz | I don't think there is a rigorous derivation. For one thing, it is possible to construct systems in which the physically-correct time evolution does not correspond to a minimization of the action. It could be a maximum of the action or—more crucially—a saddle point of the action. But what does it mean, precisely speaking, to be a saddle point of the action? Well, the fields obey the Euler-Lagrange equations.... | |
| Jan 29, 2021 at 21:33 | comment | added | Willie Wong | Most of my books are in my office; hopefully Igor Khavkine sees your question since I know for sure he has an answer to it. The correct formulation is also described in his paper arxiv.org/abs/1210.0802 but I don't know how much you would like the Jet language. I am also pretty sure that this is explained in Christodoulou's Action Principle and PDEs (Princeton Univ. Press). But I don't have my copy with me to give you a precise page reference. | |
| Jan 29, 2021 at 13:00 | comment | added | MathMath | @WillieWong this is one reason. But also, all derivations I know come from physics books and thus there are some calculuations which seem unclear or poorly justified to me. | |
| Jan 29, 2021 at 2:09 | comment | added | Willie Wong | What is unsatisfactory with the usual derivation that makes them non-rigorous? (Specifically, are you worried about the use of the compactly support perturbations? Are you worried about the fact that $S(\phi)$ is, for most field theories, necessarily infinite?) | |
| Jan 28, 2021 at 20:10 | answer | added | Carlo Beenakker | timeline score: 2 | |
| Jan 28, 2021 at 19:21 | review | Close votes | |||
| Jan 31, 2021 at 15:27 | |||||
| S Jan 28, 2021 at 19:14 | history | suggested | gmvh |
Removed inappropriate tags
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| Jan 28, 2021 at 18:02 | review | Suggested edits | |||
| S Jan 28, 2021 at 19:14 | |||||
| Jan 28, 2021 at 17:39 | history | asked | MathMath | CC BY-SA 4.0 |