So I have the lagrangian density ## L = -\frac{1}{2} W^{\mu \nu}_i W_{\mu \nu}^i## where ##W^{\mu \nu}_i = \partial ^\mu W^\nu_i - \partial ^\nu W^\mu_i + \epsilon_i^{jk}W^\mu_j W^\nu_k## and I want to find the equations of motion.(adsbygoogle = window.adsbygoogle || []).push({});

I have gotten to the stage using the EL equations ##\partial_\sigma \left(\frac{\partial L}{\partial(\partial_\sigma W_\lambda ^r)}\right)-\frac{\partial L}{\partial W_\lambda ^r}=0## that

##\partial_\sigma\left(W^{\sigma \lambda}_r-W^{\lambda \sigma}_r\right) - \frac{1}{2}\epsilon^i_{rk}\left(W_\nu^k W^{\lambda \nu}_i+W^{\lambda \nu}_i W_{\nu}^k-W_\nu^k W^{\nu \lambda}_i - W^{\nu \lambda}_iW_{\nu}^k\right)##

So I am pretty sure I need to use commutation relations to get this stuff to cancel but I don't really know what those relations even are, I know W are non abelian and hence don't commute but I don't know what their commutation relations are or anything so any help would be appreciated. I also don't know if the W tensor is anti symmetric or not but I dont know what's the relation when you swap indices.

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# A Equations of Motion for SU2 Field (Weinberg -Salam?)

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