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Hello everyone.

I've seen the usual Euler-Lagrange equation for lagrangians that depend on a vector field and its first derivatives. In curved space the equation looks the same, you just replace the lagrangian density for {-g}

The last comment ensures there's a covariant version of Euler-Lagrange for lagrangians with dependences on first order derivatives but I may ask... what happen with dependences on second derivatives in the lagrangian? - I tried to arrive at a similar result (just changing partial derivatives for covariants) but I didn't find it and I need it for my thesis.

Guys, to be concise: Have you seen the covariant version of this equation? How can I find it?

(It is supposed to look like this https://pasteboard.co/H8AyvWS.png)

I've seen the usual Euler-Lagrange equation for lagrangians that depend on a vector field and its first derivatives. In curved space the equation looks the same, you just replace the lagrangian density for {-g}

^{½}times the lagrangian density. I noticed that you can replace partial derivatives for covariant ones and easily arrive to the same result (just take into account the formula for contraction of indices in the christoffel symbols and you get this result in no more than 2 or 3 lines).The last comment ensures there's a covariant version of Euler-Lagrange for lagrangians with dependences on first order derivatives but I may ask... what happen with dependences on second derivatives in the lagrangian? - I tried to arrive at a similar result (just changing partial derivatives for covariants) but I didn't find it and I need it for my thesis.

Guys, to be concise: Have you seen the covariant version of this equation? How can I find it?

(It is supposed to look like this https://pasteboard.co/H8AyvWS.png)

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