Lagrangian/Hamiltonian of a charged particle

[dRic2]

I know that a moving particle is subjected to its own field according to Lienard-Wiechert potentials. But is it possible to write a non-relativistic Lagrangian which, upon variation of the action, give rise to the "correct" equation of motion? If such a Lagrangian/Hamiltonian exists, then is it possible to quantize it and use it in the Schrödinger equation?

This is just a curiosity, I'm not looking for some rigorous explanations. I tried googling "Lienard-Wiechert Lagragian", but I couldn't understand most of the results. "Lagrangian of a charged particle" was an other try, but all the articles I've found talked about an external field, which is not what I was looking for.

 

[vanhees71]

This is a very complicated issue, and for point particles it's not possible to give an exact system of self-consistent dynamical equations of the electromagnetic field and a point particle. It's the notorious problem of radiation reaction. The best approximation known today is the Landau-Lifshitz equation, which is an approximation of the Abraham-Lorentz-Dirac equation which has serious problems (self-acceleration, acausal solutions).

The problem is solved within continuum mechanics. A somewhat artificial but very nice model is the motion of a Born-rigid charged body:

https://arxiv.org/abs/physics/0508031
https://arxiv.org/abs/hep-th/0702078

Note that a fully consistent dynamics of a closed system of charges and em. fields is necessarily relativistic!

 

[dRic2]

Note that a fully consistent dynamics of a closed system of charges and em. fields is necessarily relativistic!

Is there a relativistic Lagrangian that can describe this process? I mean, the Lagrangian of EM field is know, the relativistic Lagrangian of the point particle is also know. What about the interaction part ? Is there a know way to explicitly write it in relativistic therms ?

 

[vanhees71]

See the 2nd paper quoted in #2. As I said for a point particle there are notorious problems which cannot be solved since ~1910!

 

[dRic2]

Do you mean Medina R 2006 Am. J. Phys. 74 1031–1034 ?

 

[vanhees71]

No I mean what I quoted above:

R. Medina, Lagrangian of the quasi-rigid charge
https://arxiv.org/abs/hep-th/0702078

To understand the problem better, I recommend the other paper by Medina, which gives also a review on the problems with point particles:

R. Medina, Lagrangian of the quasi-rigid extended charge
https://arxiv.org/abs/physics/0508031

Of course there's also a chapter about it in Jackson's textbook on electrodynamics.

 

[dRic2]

Ah ok I though they were exclusively on rigid particles. I will have lot for sure. Thanks for the material.