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The "action" is a functional of fields and their derivatives integrated over a space-time volume. A Lagrangian is just integrated over the space dimensions.

But what is the name of the thing to be integrated?

e.g.

$$S=\int L[\phi](t) dt = \int {\cal L}[\phi](x,y,z,t)dx dy dzdt.$$

What is the name of ${\cal L}$ if it has one? e.g. for the Klein-Gordon action it might be:

$${\cal L}[\phi] = \eta^{\mu\nu}\partial_\mu \phi(x)\partial_\nu \phi(x) + m^2 \phi(x)^2.$$

It's just an expression of fields and first deriviatves. Does this have a name?

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In field theory, the quantity which is integrated over spacetime to obtain the action is usually called the Lagrangian density.

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  • $\begingroup$ But since you can create a different lagrangian density by doing integration by parts, this means two lagrangigan densities will have the same Lagrangian??? $\endgroup$ – zooby Feb 8 at 19:26
  • $\begingroup$ Two lagrangian densities which only differ by a total derivative will effectively give you the same Lagrangian, since the total derivative ends up cancelling when performing the integration due to the Euler-Lagrange equations. $\endgroup$ – Charlie Feb 8 at 20:03
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The factor you integrate over spacetime is called Lagrangian density. A similar case happens when you calculate the hamiltonian, where you can integrate a Hamiltonian density.

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