Deriving the EOM for Proca Lagrangian

[a2009]

Homework Statement

Consider the Proca Lagrangian

$$L=-\frac{1}{16\pi}F^2-\frac{1}{c}J_{\mu}A^{\mu}+\frac{M^2}{8\pi}A_{\mu}A^{\mu}$$​

in the Lorentz gauge $$\partial_{\mu}A^{\mu}=0$$

Find the equation of motion.

Homework Equations

$$F^2=F_{\mu\nu}F^{\mu\nu}$$

The Attempt at a Solution

Well, first of all I'm not quite sure how E-L should look in this case. Clearly $$F_{\mu\nu}$$ is the part that is the derivatives of the dependent variables (A).
What I have gotten to is

$$\frac{\partial L}{\partial F_{\rho \sigma}} = -\frac{F^{\rho\sigma}}{8\pi}$$
$$\frac{\partial L}{\partial A_{\sigma}}=-\frac{1}{c} J^{\sigma} + \frac{M^2}{4 \pi}A^{\sigma}$$
​

So is this the E-L for 4D special relativity?

$$\frac{\partial L}{\partial A_\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0$$​

Thanks for any help.

 

[George Jones]

I would more comfortable not taking a short-cut, i.e., I would expand the F's in term of A's, but, after substituting for the L's in your final equation, you seem to get the correct equation of motion.

 

[a2009]

Thanks so much for the quick reply.

What I'm mostly unsure about is how to formulate E-L in terms of covariant or contravariant components. Do I take the derivative with respect to $$A^\sigma$$ or $$A_\sigma$$.

I guess my question could be reformulated as:

Should E-L look like

$$\frac{\partial L}{\partial A_\sigma}-\partial^\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0$$​

or

$$\frac{\partial L}{\partial A^\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F^{\rho\sigma}} \right)=0$$​

And based on what do I choose between the two?

 

[George Jones]

I guess my question could be reformulated as:

Should E-L look like

$$\frac{\partial L}{\partial A_\sigma}-\partial^\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0$$​

or

$$\frac{\partial L}{\partial A^\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F^{\rho\sigma}} \right)=0$$​

And based on what do I choose between the two?

Neither, . In each equation, $\sigma$ is a free index in one term and a dummy summed index in the other term, which is a no-no. In terms of index placement, an upstairs index in a denominator is like a downstairs index in a numerator, and a downstairs index in a denominator is like an upstairs index in a numerator.