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## Homework Statement

Consider the Proca Lagrangian

[tex]L=-\frac{1}{16\pi}F^2-\frac{1}{c}J_{\mu}A^{\mu}+\frac{M^2}{8\pi}A_{\mu}A^{\mu}[/tex]

in the Lorentz gauge [tex]\partial_{\mu}A^{\mu}=0[/tex]

Find the equation of motion.

## Homework Equations

[tex]F^2=F_{\mu\nu}F^{\mu\nu}[/tex]

## The Attempt at a Solution

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

What I have gotten to is

[tex]\frac{\partial L}{\partial F_{\rho \sigma}} = -\frac{F^{\rho\sigma}}{8\pi}[/tex]

[tex]\frac{\partial L}{\partial A_{\sigma}}=-\frac{1}{c} J^{\sigma} + \frac{M^2}{4 \pi}A^{\sigma}[/tex]

[tex]\frac{\partial L}{\partial A_{\sigma}}=-\frac{1}{c} J^{\sigma} + \frac{M^2}{4 \pi}A^{\sigma}[/tex]

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

[tex]\frac{\partial L}{\partial A_\sigma}-\partial_\sigma \left( \frac{\partial L}{\partial F_{\rho\sigma}} \right)=0[/tex]

Thanks for any help.