Timeline for Maxwell equations as Euler-Lagrange equation without electromagnetic potential
Current License: CC BY-SA 4.0
5 events
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| Dec 28, 2020 at 8:27 | comment | added | J.G. | @makt The Dirac & Schrödinger Lagrangians (the latter Galilean-invariant) use this conjugate-instead-of-new-auxiliary technique. I imagine this has also been done with Rarita–Schwinger. The Batalin–Vilkovisky formalism is an example with novel multipliers. | |
| Dec 28, 2020 at 7:11 | comment | added | asv | @J.G. : Do you have examples of physical interest? An example I am aware of is the (first order) Dirac equation say for a free particle. Its Lagrangian has no auxiliary fields or constrains. | |
| Dec 27, 2020 at 20:51 | comment | added | J.G. | @makt Indeed, it's a common technique for making an apparently first-order PDE an ELE; often, we add a total derivative, thereby unsubtly disguising what we did (although the motive may be e.g. ensuring hermiticity). When complex fields are involved, we can often use the conjugate, adjoint etc. as the multiplier, for much the same reason as $z^\ast$ can be treated as "independent" of $z$. | |
| Dec 27, 2020 at 7:14 | comment | added | asv | Thanks. It seems however that if one allows auxiliary fields, like your $\chi_\mu,\tilde\chi_\mu$, one can write any system of PDE as EL-equations. Maxwell equations are not unique in this sense. | |
| Dec 26, 2020 at 21:50 | history | answered | AccidentalFourierTransform | CC BY-SA 4.0 |