I'm finding the field equations for a hypothetical Lagrangian with dependence on the second derivative of a scalar field, $L\left(\phi,\phi_{,\mu},\phi_{,\mu\nu}\right)$, and in the analogue to the Euler-Lagrange equations I got a term that looks like $\partial_{\mu}\partial_{\nu}\frac{\partial L}{\partial \phi_{,\mu\nu}}$.

I'm still getting used to index notation -- Is $\partial_\mu\partial_\nu$ equivalent to $\partial_{\mu\nu}$?

If it is, is $\partial_{\mu\nu}\partial^{\mu\nu}\phi$ equivalent to $\square\square\phi$? (where $\square$ is the d'Alembertian)

  • 1
    $\begingroup$ Yes both of your statements are correct $\endgroup$ – Ali Moh Apr 7 '15 at 9:15

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