The Lagrangian density for a massless scalar field is $ \mathcal{L} = \frac{1}{2} ( \partial_\mu \phi ) ( \partial^\mu \phi ) $. In order to derive the equations of motion for such a Lagrangian, we plug it into the Euler-Lagrange equation. In particular we need to compute the quantity $ \partial_\mu \frac{ \partial \mathcal{L} }{ \partial( \partial_\mu \phi ) } $. I have two questions:
- Is the following derivation correct (ignoring the $1/2$) :
$$ \begin{align*} % \partial_\mu \frac{ \partial}{ \partial( \partial_\mu \phi ) } [ ( \partial_\nu \phi ) ( \partial^\nu \phi ) ] % &= \partial_\mu \frac{ \partial}{ \partial( \partial_\mu \phi ) } [ ( \partial_\nu \phi ) g^{\nu\alpha} ( \partial_\alpha \phi ) ] % \\ &= \partial_\mu g^{ \nu\alpha } [ ( \partial_\alpha \phi )\delta_\nu^\mu + ( \partial_\nu \phi ) \delta_\alpha^\mu ] \quad \text{(Using product rule)} % \\ &= \partial_\mu [ (\partial^\nu \phi ) \delta_\nu^\mu + ( \partial^\alpha \phi ) \delta_\alpha^\mu ] \quad \text{(Evaluating metric on both terms)} % \\ &= 2 \partial_\mu \partial^\mu \phi \quad \text{(Evaluating delta sum)} % \end{align*} $$
- If it was correct, was such a derivation necessary, or is there a shorter way? Using the notation $ \mathcal{L} = ( \partial_\mu \phi )^2 $ makes it look very tempting to just apply the "power rule" to get $2 \partial_\mu \phi $ when we take $ \partial / \partial ( \partial_\mu \phi) $, but it just doesn't feel right... is that legitimate math?