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0answers
45 views

Energy-momentum tensor of the electromagnetic field

I have to derive the electromagnetic energy-momentum tensor from Noether's theorem and translation invariance. Due to translation invariance and gauge transformation: $$\delta A_\mu= a^\nu F_{\mu\nu}$$...
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1answer
70 views

Invariance of Maxwell action

I have to show that the Maxwell action $$S=-\frac{1}{4}\int d^4x F^{\mu\nu}F_{\mu\nu}\,$$ is invariant under translation: $\delta_aA_\mu=a^\nu \partial_\nu A^\mu$ with $a^\mu$ as arbitrary and ...
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0answers
18 views

Constants of motion of an electron in a harmonic electromagnetic field in free space

I have encountered a question in Classical Electrodynamics, as below: In free space, an electron, initially at rest at $z=0$, is subjected to an intense laser field $\vec E=\hat x A \cos(\omega t-...
3
votes
1answer
401 views

Applying the Euler-Lagrange equations to Maxwell's Theory

In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\...
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vote
1answer
77 views

On the “Derivation of the Electromagnetic Lagrangian density”

In the most upvoted answer here : Deriving Lagrangian density for electromagnetic field, how do we know that equations (015) and (016) therein \begin{equation} \boxed{\: \dfrac{\partial }{\partial t}\...
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vote
3answers
97 views

Is $F^{\mu\nu}F_{\mu\nu}$ equivalent to $A^{\mu}\nabla^{\alpha}\nabla_{\alpha}A_{\mu}$ for $U(1)$ gauge field lagrangian?

The two seem to yield the same equation of motion is why I asked. Where of course the standard notation for exterior forms applies $dA=F$. We all know how the field strength tensor plays into the ...
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0answers
56 views

Lagrangian of Charged Particle Evaluated On-Shell

I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell. Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge ...
2
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1answer
92 views

Deriving all of electrodynamics from one single action

Preamble The action for a relativistic particle of charge $m$ and charge $q$ moving in an external electromagnetic 4-potential $A^\mu$ is $$\mathcal S_p[y]=-\int_{a}^{b}\left(mc+\frac q c A_\mu(y)\...
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0answers
830 views

Lagrangian of charged particle in magnetic field

I am aware that this question has been asked before, but the answer uses a formula I haven't seen before, and I was wondering if there is another more intuitive way to solve this problem. I wish to ...
0
votes
2answers
743 views

Hamiltonian of a charged particle in a magnetic field coincides with that of a free particle?

The Hamiltonian of a charged particle of charge $q$ in an arbitrary magnetic field $\textbf{B}$ is given by $$H=\frac{(\textbf{p}-q\textbf{A})^2}{2m}\tag{1}$$ where $\textbf{p}$ is the canonical ...
2
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0answers
36 views

Relevance of Gauge Transformations in Physical Interpretations of a System

In the simple example of a stationary electric field (and some other quantum mechanical examples) it is shown in the papers https://arxiv.org/pdf/physics/0506203.pdf https://arxiv.org/pdf/1302.1212....
2
votes
1answer
69 views

How to prove this matrix differential for Born-Infeld theory?

Consider the Born-Infeld Lagrangian, page 30 of Born-Infeld Action and Its Applications by Cong Wang. $L_{BI} = \sqrt{\det (1+ F)}$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. I ...
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vote
1answer
310 views

Lorentz force with Lagrangian

I want to prove that $$ \vec{F}=d\vec{p}/dt=q\vec{E}+(q/c) \cdot v\times \vec{B} $$ in CGS system, using $$ L=-mc^{2}/\gamma-q\phi+(q/c)\cdot \vec{v}\cdot \vec{A} \hspace{10mm} \tag 1 $$ and $$ \...
2
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0answers
258 views

Deriving the classical electromagnetic point charge Lagrangian from the Abelian Yang-Mills Lagrangian density

Can I derive the point charge Lagrangian $$ L = -\frac{mc^2}{\gamma} - \frac{1}{c} J_{\mu} A'^{\mu}\tag{1} $$ from the Abelian Yang-Mills Lagrangian $$L = \int d^3x [ - \frac{1}{16 \pi} F_{\mu ...
4
votes
1answer
305 views

Lagrangian density for Lorentz force of continuous charge distribution in external field?

It's frequently an exercise to derive the Lorentz force law for a particle with charge $q$ in an external electromagnetic field given by the following Lagrangian: $$L = -mc^2\sqrt{1-\frac{\dot{r}^2}{...
0
votes
1answer
293 views

Correct Definition of Angular Momentum of a Charged Particle in an Electromagnetic Field? (Classical Mechanics) [duplicate]

What is the more correct definition of angular momentum $\vec{\mathbf{M}}$ in three dimensions? (I.e. classically/Lagrangian/Hamiltonian, not necessarily quantum or relativistic) $$\vec{\mathbf{M}}=...
2
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0answers
75 views

Help understanding electromagnetism integral from exercise in MTW? [closed]

I was skimming through Misner, Thorne and Wheeler's book Gravitation looking for exercises to challenge myself with and came across the following exercise on page 178: Verify that the variational ...
2
votes
2answers
635 views

Why does the classical electrodynamics Lagrangian density equation have a “field” term and an “interaction” term?

On Wikipedia's page on classical electrodynamics, they state the Lagrangian density equation as follows \begin{equation} \mathcal{L} = \mathcal{L}_{\text{field}} + \mathcal{L}_{\text{int}} = -\frac{...
3
votes
1answer
678 views

Lagrangian of Non-Relativistic Charged Particle in a Magnetic Field

I'm trying to derive the Lagrangian for a non-relativistic charged particle under the influence of a magnetic potential. I'm assuming that $F=-\textrm{grad}(V)$ and so by the Lorentz force we have $-\...
10
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1answer
1k views

Recovering all of Maxwell's equations from the variational principle

Whether you can get the first couple of Maxwell equations from a variational principle? In the second volume of the Landau theoretical physics said that it is impossible.
2
votes
1answer
846 views

Canonical Stress Tensor for the Free Electromagnetic Field

I have the followwing Lagrangian for the free electromagnetic field, $$\mathcal{L} = -\frac{1}{4} F^{\mu \nu}F_{\mu \nu},$$ and the canonical stress tensor is, $$T^{\alpha \beta}=\frac{\partial \...
14
votes
1answer
934 views

Physical Interpretation of EM Field Lagrangian

Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density? The ...
3
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2answers
2k views

Lorentz force equation from relativistic Lagrangian

The relativistic Lagrangian is given by $$L = - m_0 c^2 \sqrt{1 - \frac{u^2}{c^2}} + \frac{q}{c} (\vec u \cdot \vec A) - q \Phi $$ I need to derive, $\displaystyle \frac{d\vec p}{dt} = q \left( \vec E ...
5
votes
1answer
672 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
12
votes
3answers
6k views

Maxwells Equation from Electromagnetic Lagrangian

In Heaviside-Lorentz units the Maxwell's equations are: $$\nabla \cdot \vec{E} = \rho $$ $$ \nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$ $$ \nabla \times \vec{E} + \frac{\...
2
votes
1answer
347 views

Question on 1st order Lagrangian Derivation in Faddeev-Jackiw Formalism

I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure). The topic is the Faddeev-Jackiw treatment of ...
6
votes
5answers
2k views

What is the Lagrangian for a relativistic charge that includes the self-force?

The usual Lagrangian for a relativistically moving charge, as found in most text books, doesn't take into account the self force from it radiating EM energy. So what is the Lagrangian for a ...