- #1

- 157

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Can anyone explain why from varying a Lagrangian?

[itex] \frac{\partial \mathcal{L}}{\partial (\partial_{\alpha }\phi)}(\frac{1}{2}\sqrt{-g}g^{\alpha\beta}\partial_{\alpha} \phi \partial_{\beta} \phi) = g^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi + \frac{1}{\sqrt{-g}}(\partial_{\nu}\phi)\partial_{\mu}(g^{\alpha \beta}\sqrt{-g}) [/itex]

I realise that the indices are dummies and so can be renamed accordingly but I don't understand why it isn't equal to:

[itex] (\frac{1}{2}g^{\mu\nu} \partial_{\nu} \phi) + \frac{1}{\sqrt{-g}}(\partial_{\mu} \phi) \partial_{\mu}(\frac{1}{2}\sqrt{-g}g^{\mu\nu} \partial_{\nu} \phi)

[/itex]