Electromagnetic Tensor - Lagrangian Formulation of Classical Electromagnetism (no Charges and Currents)

Lagrangian Formulation of Classical Electromagnetism (no Charges and Currents)

See also: Classical field theory

When there are no electric charges (ρ = 0) and no electric currents (J = 0), Classical electromagnetism and Maxwell's equations can be derived from the action:

where

is over space and time.

This means the Lagrangian density is

\begin{align} \mathcal{L} & = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} \\ & = - \frac{1}{4\mu_0} \left( \partial_\mu A_\nu - \partial_\nu A_\mu \right) \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) \\ & = -\frac{1}{4\mu_0} \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu - \partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\nu A^\mu \right)\\ \end{align}

The two middle terms are also the same, so the Lagrangian density is

Substituting this into the Euler-Lagrange equation of motion for a field:

The second term is zero because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:

The quantity in parentheses above is just the field tensor, so this finally simplifies to

That equation is another way of writing the two inhomogeneous Maxwell's equations, making the substitutions:

where i, j take the values 1, 2, and 3.

When there are charges or currents, the Lagrangian needs an extra term to account for the coupling between them and the electromagnetic field. In that case is equal to the 4-current instead of zero.

Read more about this topic:  Electromagnetic Tensor

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