TITLE: Lagrangian formalism in electrodynamics
QUESTION [0 upvotes]: 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.

REPLY [1 votes]: 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.
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