In Classical Mechanics one usually considers the Lagrangian as $L = K - U$ where $K$ is the kinetic energy of the system and $U$ is the potential energy. One then gets the Euler-Lagrange equations and everything is fine: if we have a system we can plug in the kinetic energy and potential and find the Lagrangian for it.

The point is that I've already seen a different object: the Lagrangian density $\mathcal{L}$ which is on the other hand a $4$-form on space-time. The main difference being that the action is the integral of $L$ over time and the integral of $\mathcal{L}$ over all space-time.

The problem is that apart from that, no relation between $\mathcal{L}$ and other quantities is given at first. So for instance, in electrodynamics we have

$$\mathcal{L} = -\dfrac{1}{4\mu_0}F^{\alpha \beta}F_{\alpha\beta}-A_\alpha J^\alpha$$

Where $A$ is the $4$-potential and $F=dA$ is the electromagnetic tensor. It is not clear at first, why this is the right Lagrangian density in the sense that it becomes a little hard to see where it comes from.

So, the Lagrangian itself is just $K-U$, but what about the Lagrangian density? How does one find it?


The easiest answer is "because that generates Maxwell's equations".

The slightly more difficult answer is that the Lagrangian density has to be gauge invariant and a Lorentz scalar. The objects we have at hand are $F_{\alpha\beta}$, $A_\alpha$ and $J^\alpha$. Now, something like $A^\alpha A_\alpha$ is out, because it is not gauge invariant, $A^\alpha A^\beta F_{\alpha\beta}$ and $J^\alpha J^\beta F_{\alpha\beta}$ are 0 (symmetric tensor contracted with an antisymmetric), $J^\alpha A^\beta F_{\alpha\beta}$ is not gauge invariant. $F_{\alpha\beta}F^{\alpha\beta}$ on the other hand is gauge invariant (because $F_{\alpha\beta}$ is). $A_\alpha J^\alpha$ is not gauge invariant, but the extra term is a total divergence, so the action $S = \int d^4 \mathcal L$ will be gauge invariant.

So $$\mathcal L = aF^{\alpha\beta}F_{\alpha\beta} - bA_\alpha J^\alpha$$ for some constants $a, b$ is the simplest interacting Lagrangian density. You can fix the constants by matching them to the ones in Maxwell's equations.


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