Derivation of the Eqation of Motion from Fermi Lagrangian density

[radioactive8]

Homework Statement

Hello, I am trying to find the equations of motion that come from the fermi lagrangian density of the covariant formalism of Electeomagnetism.

Homework Equations

The form of the L. density is:
$$L=-\frac{1}{2} (\partial_n A_m)(\partial^n A^m) - \frac{1}{c} J_m A^m$$

where J is the electric current.
The result has to be:

$$\partial_n \partial^n A^m = \frac{1}{c} J^m$$

The Attempt at a Solution

Using the Euler- Lagrange equations that derive the eq. of motion I do not understant if I have to treat the fields A^m and A_m. In addition, the Euler Lagrange equations from a L.density of the form:
$$L=-\frac{1}{2} (\partial_n A_m)(\partial_n A_m) - \frac{1}{c} J_m A^m$$
giveaway the wanted result. But I could not relate the two L.densities.

 

[maajdl]

How did you lower the indices to get the density in §3 ?
I think the density in §3 is not covariant and it is then incorrect.
There must be elements of the metric tensor missing in the density of §3 .

 

[Orodruin]

How did you lower the indices to get the density in §3 ?
I think the density in §3 is not covariant and it is then incorrect.
There must be elements of the metric tensor missing in the density of §3 .

Indeed. OP, please show your work.

 

[radioactive8]

I did not lower any indices. And yes it is not covariant at all so it could not be a lagrangian. It was something a digged up on the internet while I was working at the solution of my problem. However, I did not notice while posting the thread that it was not covariant.

Now, Let's focus on the first L.density.

I tried playing with the indexes and my metric.

Is the following correct? :

$$L=-1/2(\partial_n A_m)( \partial^n A^m) = -\frac{1}{2} (g_{ns} \partial^s A_m)(g^{nr} \partial_r A^m) = -\frac{1}{2} \delta_{s}^{r}(\partial^s A_m)( \partial_r A^m)= -\frac{1}{2} (\partial^s A_m)(\partial_s A^m) $$

So the partial derivative $$\frac{ \partial L}{\partial (\partial_s A^m)}$$ equals:

$$\partial_s (\frac{ \partial L}{\partial (\partial_s A^m)})= \partial_s \partial^s A^m$$