Solving Lorentz Condition with Lagrangian

[nicksauce]

Homework Statement

Given the Lagrangian
$$L = -\frac{1}{2}\partial_{\alpha}A_{\beta}\partial^{\alpha}A^{\beta} + \frac{1}{2}\partial_{\alpha}A^{\alpha}\partial_{\beta}A^{\beta} + \frac{\mu^2}{2}A_{\beta}A^{\beta}$$

show that A satisfies the Lorentz condition $\partial_{\alpha}A^{\alpha} = 0$.

Homework Equations

The Attempt at a Solution

I want to say we can treat $\partial_{\alpha}A^{\alpha}$ as an independent field, and find the appropriate field equations for it, but I'm not sure if that makes sense. Any thoughts?

 

[nicksauce]

Upon further thought, this seems like a good time to use Noether's theorem...

 

[orentago]

Also see an alternative method https://www.physicsforums.com/showthread.php?p=2979874#post2979874".