# Lagrangian/Hamiltonian of a charged particle

• dRic2
In summary, it is not possible to provide a relativistic Lagrangian which will give the correct equation of motion for a point particle subjected to its own field.
dRic2
Gold Member
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 Schrodinger 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.

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
vanhees71 said:
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 ?

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

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

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.

## 1. What is the Lagrangian/Hamiltonian of a charged particle?

The Lagrangian and Hamiltonian are mathematical functions that describe the dynamics of a charged particle in a given system. They take into account the particle's position, velocity, and acceleration, as well as any external forces acting on it.

## 2. How are the Lagrangian and Hamiltonian related?

The Lagrangian and Hamiltonian are related through a mathematical transformation known as the Legendre transformation. The Hamiltonian is equal to the Lagrangian plus the product of the particle's momentum and velocity.

## 3. What is the significance of the Lagrangian/Hamiltonian in physics?

The Lagrangian and Hamiltonian are important in physics because they provide a complete and elegant description of a system's dynamics. They can be used to derive equations of motion and predict the behavior of a charged particle in a given system.

## 4. How do the Lagrangian/Hamiltonian equations differ from Newton's laws of motion?

The Lagrangian and Hamiltonian equations differ from Newton's laws of motion in that they are based on the principle of least action, which states that a system will follow the path that minimizes the action (a measure of energy) over time. This approach allows for a more general and powerful description of a system's dynamics.

## 5. Can the Lagrangian/Hamiltonian equations be applied to systems with multiple charged particles?

Yes, the Lagrangian and Hamiltonian equations can be applied to systems with multiple charged particles. In these cases, the equations become more complex and involve the interactions between the particles, but they still provide a comprehensive description of the system's dynamics.

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