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I know formalism of Euler_Lagrange derivatives:

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which $L$ here is a Lagrangian density.

now I wonder how can I make such a form as below for any Lagrangian density which is a function of metric and its first and second derivatives?

enter image description here

My main problem is transforming the first two terms of Euler_Lagrange derivatives formalism to this shape.

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  • $\begingroup$ There is a cool derivation of almost the same in Carroll's book on GR... $\endgroup$ – user192234 Jul 19 at 9:46

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