Timeline for Maxwell equations as Euler-Lagrange equation without electromagnetic potential
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2020 at 20:37 | comment | added | Terry Tao | Details are now given in a separate answer to this question. | |
| Dec 28, 2020 at 16:56 | comment | added | Terry Tao | p.s. linearised Yang-Mills is another excellent analogy for Maxwell; expressible by fluke in terms of the curvature 2-form, but most naturally thought of using the connection 1-form (and in particular has a Lagrangian formulation in terms of the Yang-Mills functional subject to the constraint of being the curvature of such a form, but no natural Lagrangian as an unconstrained 2-form). Indeed Maxwell is Yang-Mills for the abelian gauge group U(1), which is automatically linear due to the abelian nature of the group. | |
| Dec 28, 2020 at 16:41 | comment | added | Terry Tao | However, such a mindset would have hindered the ability to discover the full Einstein equations, or to discover more general equations of motion (e.g., Einstein-Maxwell, Einstein-Vlasov, etc.) in which gravitation is coupled with other physical fields. As I said, the ability for Maxwell's equations to be expressible purely in terms of the curvature tensor $F_{\alpha \beta}$ (analogous to $R_{\alpha \beta \gamma}^\delta$) is a fluke (only shared by a few other linear field models such as linearised gravity) and is somewhat misleading as a guide to more general physical field theories. | |
| Dec 28, 2020 at 16:38 | comment | added | Terry Tao | One can imagine as a thought experiment that the equations for linearised gravity (Bianchi+Einstein, expressed in terms of a tensor field $R_{\alpha \beta \gamma}^\delta$) were discovered before the full Einstein equations. It was observed that it was technically convenient to express this tensor $R_{\alpha \beta \gamma}^\delta$ as the Riemann curvature of a "metric potential" $h_{\alpha \beta}$, but the equations could be written without reference to such a potential, so one might be led to the belief that $R_{\alpha \beta \gamma}^\delta$ could be treated as an unconstrained tensor. | |
| Dec 28, 2020 at 16:34 | comment | added | Terry Tao | Admittedly the nonlinear aspects of general relativity weaken the analogy with the linear equations of Maxwellian electromagnetism. But one can easily repair this aspect of the analogy by coimparing Maxwell to linearised gravity (en.wikipedia.org/wiki/Linearized_gravity) instead. Now the Bianchi identities use the Minkowski metric instead of a variable coefficient metric, and similarly the Einstein-Hilbert functional will use the Minkowski volume element instead of one generated by a dynamic metric. | |
| Dec 28, 2020 at 6:39 | comment | added | asv | @TerryTao: May I also add that the formulation of Bianchi identity requires the metric (Christoffel symbols), it is not formulated purely in terms of the Riemann curvature tensor. | |
| Dec 28, 2020 at 5:41 | comment | added | Michael Engelhardt | @makt - in our current context, $\nabla \cdot B=0$ and $\nabla \times E = -\dot{B} $ comprise the Bianchi identities ... | |
| Dec 28, 2020 at 5:31 | comment | added | asv | @TerryTao: As far as I understand, in GR the curvature tensor is not enough to write Hilbert action $\int s_g dvol_g$: one has to use the metric $g$ to write the volume form. Bianchi identities are neither constrains nor equations of motions, in my understanding. To get EL equation one varies the metric rather then the Riemann curvature tensor constrained by Bianchi identities. | |
| Dec 28, 2020 at 2:23 | comment | added | Terry Tao | Ultimately the apparent symmetry between the Gauss-Faraday and Gauss-Ampere components of Maxwell's equation, and the apparent fundamental nature of the electromagnetic field, is a fluke of classical electromagnetism that is not replicated in more sophisticated electromagnetic theories, in which it really is the vector potential (or, as I like to think of it, the connection on the electromagnetic U(1) bundle) which is the underlying physical field. This can be seen either in quantum mechanical electromagnetism (Bohm-Aharonov effect) or in classical coupled systems like Maxwell-Klein-Gordon. | |
| Dec 28, 2020 at 2:16 | comment | added | Terry Tao | @makt "I have never seen it in other examples of constructing Lagrangians in classical mechanics"... actually general relativity provides a good example. If one wished to formulate GR in terms of the Riemann curvature tensor instead of the metric (much as Maxwell's equations are formulated in terms of the EM field rather than the vector potential), one would see a "mysterious and artificial" division into the "constraint" equations (Bianchi identities) and "dynamic" equations (Einstein), with only the latter having a Lagrangian interpretation (Einstein-Hilbert action). | |
| Dec 27, 2020 at 17:28 | comment | added | Michael Engelhardt | ... this would also match Terry Tao's comment that at most 6 of Maxwell's equations can be obtained purely from a formulation in terms of $\vec{E} $ and $\vec{B} $ fields - meaning we need 2 explicit extra Lagrange multipliers to enforce the other 2 equations. The choices we make in organizing the fields and equations are suggested by the symmetries, most importantly, Lorentz invariance. So, I'd say that's more than just technical - but you are right, these are choices, and we could choose to express electromagnetism in much more convoluted ways. | |
| Dec 27, 2020 at 17:23 | comment | added | Michael Engelhardt | Well there really are 2 degrees of freedom, so we can't view all of Maxwell's equations as constraints. Two of them must be dynamical. So the question is, how do we count the rest. Now, for each constraint, there should be 2 equations - one for the constrained field and one for the Lagrange multiplier. Now, like in the vector potential case, some of the fields may already act as Lagrange multipliers! Just by counting, I suspect that we only actually only need 2 new multipliers, whose E-L equations bring us to a total of 10 equations, 4+4 of which implement constraints, leaving 2 dynamical ... | |
| Dec 27, 2020 at 16:52 | comment | added | asv | Probably I start seeing your point. But in that case why do not we consider all Maxwell equaitons as constrains and no dynamics equations? Why do not we choose any other subset of Maxwell equations as constrains? Is it just technical convenience? I have never seen that in literature. | |
| Dec 27, 2020 at 16:36 | comment | added | Michael Engelhardt | ... and we do readily distinguish the corresponding Euler-Lagrange equations into dynamical and constraint equations. | |
| Dec 27, 2020 at 16:35 | comment | added | Michael Engelhardt | Well, I would on the contrary say that, from the physical point of view, the electromagnetic field has 2 degrees of freedom. That is what is actually seen in electromagnetic waves! We're the ones who are inflating the number of components to 6. Of course, we have reasons to do so, related to the different ways we are able to observe electromagnetic phenomena. I don't think this is that exceptional - think of a mechanical object constrained to move on a circle of radius $R$ in a plane. We can and do choose at times to consider that in terms of variables $r$, $\phi $ with a constraint $r=R$ ... | |
| Dec 27, 2020 at 16:20 | comment | added | asv | Formally I understand the argument, but I would like to understand if one can avoid it. It looks really ad hoc. I do not see any reason, other then the technical one, to distinguish a part of Maxwell equations from the rest. | |
| Dec 27, 2020 at 16:20 | comment | added | asv | Thanks very much. The answer contains the same point I am trying to avoid. From the physical point of view electromagnetic field has 6 degrees of freedom $(\vec E,\vec B)$. When you assume existence of potential $A^\mu$ you assume half of the Maxwell equations. So you consider them as constrains rather than equations of dynamics. The other half of Maxwell equaitons are considered as equations of dynamics. This division of Maxwell equations into two types seems to me to be mysterious and artificial. I have never seen it in other examples of constructing Lagrangians in classical mechanics. | |
| Dec 27, 2020 at 16:00 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |