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In classical mechanics we talk about Lagrangian, but when we talk about fields (in example- electromagnetic fields), we consider the Lagrangian density instead of "just" the Lagrangian. I didn't understand why? In addition I didn't understand the differences between the relativistic Lagrangian formalism and classical Lagrangian formalism when we are considering fields.

Qmechanic
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BERTI2020
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  • For a point mass in rigid body mechanics, one usually writes an energy function $L=L(x,\dot x)$ describing the mass whose position,velocity are $x$,$\dot x$. For a field, there's finite extent and the single position $x$ and its derivatives are insufficient to describe a field covering more than one point. It won't do to simply write the total energy because then you have no coordinate dependence with which to derive the equations of motion. For this reason, we write the energy function as integral over the energy density (which gives the total energy.) This preserves the coordinate dependence – hodop smith Oct 31 '20 at 20:19
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    Does this answer your question? Why do we consider Lagrangian densities in QFT? – CuriousHegemon Oct 31 '20 at 20:46
  • YES! Thanks a lot! – BERTI2020 Oct 31 '20 at 22:44

2 Answers2

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It is the difference between a discrete and a continuum approach.

In classical discrete mechanics, there are particles moving in 3D space along the time. The action is the sum of all integrals of their Lagrangians between $t_1$ to $t_2$.

But in continuum classical mechanics for example, the displacement field of the material have to be dealt with, and Lagrange density is used. So, even in classical mechanics this concept can be applied.

In the case of relativity, the difference is that the time coordinate is not independent of the spacial ones. But the Lagrangian density have to be integrated in space and time as in continuum mechanics.

Claudio Saspinski
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I'm not an expert, but I believe it's simply more convenient to talk about the Lagrangian density since you can then talk about integrals over a region in spacetime as opposed to integrals over a time interval, which fits better into the framework of covariant notation.

Rindler98
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