The function that fully describes a physical system in Lagrangian mechanics.

When we particles particles, the action is obtained by integrating the Lagrangian over time:

$Action=∫_{t_{0}}L(x(t),∂t∂x(t) ,t)dt$

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field $f(x,y,t)$:

$Action=∫_{t_{0}}L(f(x,y,t),∂x∂f(x,y,t) ,∂y∂f(x,y,t) ,∂t∂f(x,y,t) ,x,y,t)dxdydt$

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with $x∈R_{2}$:

$Action=∫_{t_{0}}L(f(x,t),∇f(x,t),x,t)dxdt$

And in the context of special relativity, people condense that even further by adding $t$ to the spacetime Four-vector as well, so you don't even need to write that separate pesky $t$.

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.

The variables of the Lagrangian, e.g. the angles of a double pendulum. From that example it is clear that these variables don't need to be simple things like cartesian coordinates or polar coordinates (although these tend to be the overwhelming majority of simple case encountered): any way to describe the system is perfectly valid.

In quantum field theory, those variables are actually fields.

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