hello, everyone. When a vector field ##A_{\mu}## has the Lagrangian of the form as(adsbygoogle = window.adsbygoogle || []).push({});

##L=Const.{\times}F^{\mu\nu}F_{\mu\nu}##, where

[itex]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}[/itex]. Now I will apply the least action principle to it. Which one of the following two choices is the right one?

[itex]{\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,[/itex]

or

[itex]{\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.[/itex]

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# A About the least action principle

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