What is the equation of motion for the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G \partial\_\mu G^{-1})$$where $G$ is an $N \times N$ unitary matrix? Could anyone supply a reference to its computation?



Equation of motion for the Lagrangian $\mathcal{L} = \text{Tr}(\partial^\mu G \partial_\mu G^{-1})$, $G$ is unitary $N \times N$ matrix?
The variation of the Lagrangian is fairly straight forward, keeping in mind the identity $\delta G^{-1} = - G^{-1} (\delta G) G^{-1}$ and the fact that $\mathrm{Tr}(AB)$ is a non-degenerate bilinear form.
\begin{align\*}
 \delta\mathcal{L}
 &= \mathrm{Tr}(\partial\_\mu \delta G \partial^\mu G^{-1})
 - \mathrm{Tr}(\partial\_\mu G \partial^\mu (G^{-1} \delta G G^{-1})) \\
 &= -\mathrm{Tr}(\delta G \partial\_\mu \partial^\mu G^{-1})
 + \mathrm{Tr}(\delta G G^{-1} (\partial\_\mu \partial^\mu G) G^{-1})
 + \partial\_\mu (\cdots)^\mu
\end{align\*}
So, the corresponding Euler-Lagrange equations are
$$ -\partial\_\mu \partial^\mu G^{-1} + G^{-1} (\partial\_\mu \partial^\mu G) G^{-1} = 0 . $$
Using the chain rule one can rewrite this in a way where the highest order derivatives act only on $G$,
$$ 2 (\partial\_\mu \partial^\mu G) + G (\partial\_\mu G^{-1})(\partial^\mu G) + (\partial^\mu G) (\partial\_\mu G^{-1}) G = 0 , $$
or where all derivatives act only on $G$,
$$ 2 (\partial\_\mu \partial^\mu G) - 2(\partial\_\mu G) G^{-1} (\partial^\mu G) = 0 . $$


