Timeline for Maxwell equations as Euler-Lagrange equation without electromagnetic potential
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2022 at 21:04 | answer | added | Maxwell | timeline score: 2 | |
| Dec 28, 2020 at 18:41 | answer | added | Terry Tao | timeline score: 9 | |
| Dec 27, 2020 at 19:05 | answer | added | Terry Tao | timeline score: 10 | |
| Dec 27, 2020 at 18:23 | comment | added | Terry Tao | ... actually, I withdraw this belief; since we are working over the reals, one could combine the eight Maxwell's equations into a single equation by taking a sum of squares of each of the eight equations. So some other means would be needed to definitively rule out Maxwell's equations having an unconstrained EL formulation. | |
| Dec 27, 2020 at 18:16 | comment | added | Terry Tao | Basically there is a differential algebra question to be answered here: what is the minimal number of generators for the differential ideal generated by the Maxwell equations (viewed as an ideal in the differential algebra generated by $E,B$ and its derivatives?). The equations themselves form an eight-element set of the ideal, and this I believe is minimal, but proving this requires some non-trivial differential algebra (the ideal does not appear to be a "complete intersection", so naive degree of freedom counting doesn't give adequate bounds). | |
| Dec 27, 2020 at 18:13 | comment | added | asv | @TerryTao: " one cannot get Maxwell on the nose from unconstrained EL." This is what my original question was about. | |
| Dec 27, 2020 at 18:08 | comment | added | Terry Tao | To specify the EL equations one needs to select both a Lagrangian and the set of constraints (the latter giving Lagrange multiplier terms in the EL equations). As you say, standard Lagrangian plus no constraints gives too strong a set of equations; but standard Lagrangian plus the constraint of being closed (or a curvature) recovers the Maxwell equations. I'm pretty sure the Maxwell equations can't be generated from 6 or fewer differential equations (though formally proving this would require some nontrivial differential algebra), so one cannot get Maxwell on the nose from unconstrained EL. | |
| Dec 27, 2020 at 16:00 | answer | added | Michael Engelhardt | timeline score: 2 | |
| Dec 27, 2020 at 7:19 | history | edited | asv | CC BY-SA 4.0 |
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| Dec 27, 2020 at 7:09 | history | edited | asv | CC BY-SA 4.0 |
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| Dec 27, 2020 at 6:51 | comment | added | asv | @TerryTao: Your first sentence sounds to be in the right direction. However I do not understand the following example. If one considers the standard Lagrangian $\frac{1}{2}\int F^{\mu\nu}F_{\mu\nu}$ where $F_{\mu\nu}$ is any (not necessarily closed) 2-form, then the EL-equation is 6 equations $\vec E=\vec B=0$ (equivalently $F_{\mu\nu}=0$). It is much stronger than 8 Maxwell equations. | |
| Dec 27, 2020 at 6:44 | comment | added | asv | @KonstantinosKanakoglou: In this post I explicitly suppose that there are no charges, i.e. no term $j_{\alpha}A^{\alpha}$. | |
| Dec 27, 2020 at 4:15 | comment | added | Terry Tao | If you are going to think of the electromagnetic field $F_{\alpha \beta}$ as an arbitrary two-form (as opposed to the curvature of a connection one-form $A_\alpha$), without any other additional fields present, then any Euler-Lagrange equation can only achieve six out of the eight Maxwell equations at best. If however you constrain $F_{\alpha \beta}$ to be a curvature, then the Yang-Mills Lagrangian $\frac{1}{2} \int F^{\alpha \beta} F_{\alpha \beta}$ works just fine, providing the four of the eight Maxwell equations not coming from curvature forms being closed. | |
| S Dec 27, 2020 at 1:48 | history | suggested | AccidentalFourierTransform | CC BY-SA 4.0 |
fixin stuff
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| Dec 26, 2020 at 21:57 | review | Suggested edits | |||
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| Dec 26, 2020 at 21:53 | comment | added | AccidentalFourierTransform | x-posted on physics.SE: physics.stackexchange.com/q/603014/84967 | |
| Dec 26, 2020 at 21:50 | answer | added | AccidentalFourierTransform | timeline score: 6 | |
| Dec 26, 2020 at 20:09 | comment | added | Konstantinos Kanakoglou | I suppose you are aware of Jackson's book on classical electrodynamics and the Lagrangian density relation (12.85), p.599, 3rd edition ? And you are not satisfied by this, due to the the second summand $\frac{1}{c}j_\alpha A^\alpha$ ? | |
| Dec 26, 2020 at 18:54 | history | became hot network question | |||
| Dec 26, 2020 at 15:54 | history | edited | asv | CC BY-SA 4.0 |
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| Dec 26, 2020 at 11:25 | answer | added | Denis Serre | timeline score: 13 | |
| Dec 26, 2020 at 11:17 | history | edited | asv | CC BY-SA 4.0 |
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| S Dec 26, 2020 at 10:58 | history | suggested | cngzz1 | CC BY-SA 4.0 |
fixed grammar.
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| Dec 26, 2020 at 10:56 | review | Suggested edits | |||
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| Dec 26, 2020 at 10:54 | history | asked | asv | CC BY-SA 4.0 |