= Adjoint Yang-Mills-Higgs trace variation
{title2=$D_\mu F^{\mu\nu}=[D^\nu\Phi,\Phi],\quad D_\mu D^\mu\Phi+m^2\Phi=0$}
Take $D_\mu X=\partial_\mu X+[A_\mu,X]$, signature $(+---)$ and the <trace> <Lagrangian density> $\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})/4-\operatorname{Tr}(D_\mu\Phi D^\mu\Phi)/2+m^2\operatorname{Tr}(\Phi^2)/2$. The variation of the <gauge curvature> is $D_\mu\delta A_\nu-D_\nu\delta A_\mu$. Cyclicity of the <trace> and covariant <integration by parts> give gauge coefficient $-D_\mu F^{\mu\nu}+[D^\nu\Phi,\Phi]$ and scalar coefficient $D_\mu D^\mu\Phi+m^2\Phi$. Their vanishing gives the displayed <Euler-Lagrange field equations>. In static <temporal gauge> at zero mass they reduce to $D_jF_{ji}=[\Phi,D_i\Phi]$ and $D_iD_i\Phi=0$.
// Destination: integrable-systems.bigb
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