Electromagnetic Lagrangian, EoM, Polarisation States

This means that ##F^{00} = 0## according to the definition of ##F^{\mu \nu}##.In summary, the conversation pertains to a question about the Euler-Lagrange equations and finding the equation of motion. The solution provided involves reducing the sum over ##\mu## to only the ##\mu=i## terms and it is questioned why there are no terms for ##\mu=0##. It is then explained that for an antisymmetric tensor, all diagonal entries must be zero, including ##F_{00}##. This leads to the conclusion that ##F^{00}## must also be zero according to the definition of ##F^{\mu \nu}##.
  • #1
binbagsss
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11

Homework Statement



Attached:

queey.png

Homework Equations



Euler-Lagrange equations to find the EoM

The Attempt at a Solution


[/B]
Solution attached:

solly.png


I follow, up to where the sum over ##\mu## reduces to sum over ##\mu=i## only, why are there no ##\mu=0## terms? I don't understand at all.

Many thanks
 

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  • #2
According to equation (9), what can you say about ##F_{00}##? What does this tell you about ##F^{00}##?
 
  • #3
TSny said:
According to equation (9), what can you say about ##F_{00}##? What does this tell you about ##F^{00}##?

oh for a antisymmetric tensor all diagonal entries must be zero? ##F_{ii}=0 ## as is ##F_{00} ##?
 
  • #4
binbagsss said:
oh for a antisymmetric tensor all diagonal entries must be zero? ##F_{ii}=0 ## as is ##F_{00} ##?
Yes. The expression for ##F_{\mu \nu}## in (9) shows that ##F## is antisymmetric. In particular, if you let ##\mu = 0## and ##\nu = 0## in the expression for ##F_{\mu \nu}## in (9) then you see that ##F_{00} = 0##.
 

Related to Electromagnetic Lagrangian, EoM, Polarisation States

What is the Electromagnetic Lagrangian?

The Electromagnetic Lagrangian is a mathematical description of the electromagnetic field. It is a function that takes into account the time and spatial coordinates to describe the energy and momentum of the electromagnetic field. It is an important tool in understanding the behavior of electromagnetic waves and their interactions with matter.

What are the EoM in relation to Electromagnetic Lagrangian?

The EoM, or equations of motion, are equations derived from the Electromagnetic Lagrangian that describe the behavior of the electromagnetic field. They are used to determine the trajectories and evolution of the electromagnetic field over time.

What are Polarisation States in Electromagnetic Lagrangian?

Polarisation states refer to the different orientations of the electric and magnetic fields in an electromagnetic wave. These states can be linear, circular, or elliptical, and they play a crucial role in determining the properties of the electromagnetic wave.

How do the Polarisation States affect the behavior of electromagnetic waves?

The polarisation states of an electromagnetic wave determine its direction of propagation, intensity, and ability to interact with matter. They also affect the speed and frequency of the wave, as well as how it reflects and refracts when encountering different materials.

How is the Electromagnetic Lagrangian used in practical applications?

The Electromagnetic Lagrangian is used extensively in various fields such as optics, electronics, and telecommunications. It is used to study and predict the behavior of electromagnetic waves in different materials and environments, and it is also a crucial component in the development of technologies such as lasers, antennas, and optical fibers.

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