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The lagrangian for scalar field is defined as, $$L=L(\phi,\partial_{\mu}\phi,\partial_{\mu}\partial_{\nu}\phi)\tag{1}$$ $but$ there is also another lagrangian which is defined as, $$L=L(\phi,\nabla_{\mu}\phi,\nabla_{\mu}\nabla_{\nu}\phi)\tag{2}$$ what is the difference between these two lagrangian?

Is first lagrangian defined for flat spacetime and second lagrangian is defined for curved spacetime?

Keshav shrestha
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  1. The Lagrangian (2) can always be rewritten into the form of the Lagrangian (1) [with possibly a different functional dependence]. The other way requires some geometric input, i.e. a connection $\nabla$.

  2. Concerning whether to use partial or covariant derivatives in the Euler-Lagrange (EL) equations, see e.g. this related Phys.SE post.

  3. For higher-derivative theories, additional issues arises due to non-commutativity of covariant derivatives.

Qmechanic
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There might be different habits in different fields of physics (you don't specify which one you're talking about), but for me, coming from quantum field theory, $\partial_\mu$ and $\nabla_\mu$ are exactly the same thing.

Miyase
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