Least Action Principle Applied to Vector Field ##A_{\mu}##

[yancey]

hello, everyone. When a vector field $A_{\mu}$ has the Lagrangian of the form as
$L=Const.{\times}F^{\mu\nu}F_{\mu\nu}$, where
$F_{\mu\nu}=({\partial}_{\mu}-{\alpha}{\partial}^{\rho}{\partial}_{\rho}{\partial}_{\mu})A_{\nu}-({\partial}_{\nu}-{\alpha}{\partial}^{\rho}{\partial}_{\rho}{\partial}_{\nu})A_{\mu}$. Now I will apply the least action principle to it. Which one of the following two choices is the right one?
${\delta}S=\frac{{\partial}S}{{\partial}A_{\mu}}{\delta}A_{\mu}+\frac{{\partial}S}{{\partial}({\partial}_{\mu}A_{\nu})}{\delta}({\partial}_{\mu}A_{\nu})=0,$
or
${\delta}S=\frac{{\partial}S}{{\partial}A_{\mu}}{\delta}A_{\mu}+\frac{{\partial}S}{{\partial}({\partial}_{\mu}A_{\nu})}{\delta}({\partial}_{\mu}A_{\nu})+\frac{{\partial}S}{{\partial}({\partial}_{\mu}{\partial}_{\nu}{\partial}_{\rho}A_{\lambda})}{\delta}({\partial}_{\mu}{\partial}_{\nu}{\partial}_{\rho}A_{\lambda})=0.$

 

[Orodruin]

Which do you think and why?

 

[haushofer]

What is that alpha in your field strength? I'd say F=dA.

You can include higher derivatives in your functional derivative; if you use partial integrations, you'll see that every higher order derivative swaps a sign.

 

[yancey]

What is that alpha in your field strength? I'd say F=dA.

You can include higher derivatives in your functional derivative; if you use partial integrations, you'll see that every higher order derivative swaps a sign.

Thanks for your attention. I was reading a paper on a model of generalized uncertainty principle which originated from quantum gravity effect. $\alpha$ represents the parameter of quantum gravity effect.