Derivation of Maxwell equations.

[arroy_0205]

In source free case the Maxwell equation are
$$\partial_{\mu}F^{\mu \nu}=0;\,\,\partial_{\mu}\tilde{F}^{\mu \nu}=0$$
I know how to derive the first equation from action principle. But how do I derive the second one containing dual field tensor form an action principle?It seems in the textbooks the second equation is "constructed" by observing the maxwell equation in terms of E amd B and the the field tensor, and not shown how to "derive" it form a basic principle. Can anybody tell me how to do that? Actually the dual tensor and the original tensor can be expressed in terms of each other, so one can say we can rewrite the action in terms of the dual tensor and then derive the equation. But that approach is also not independent of the original tensor. I am confused about which tensor is actually fundamental? Since these give different Maxwell equations, they should be independent. How to resolve this?

 

[arroy_0205]

I found out one fact, that the equation relating the dual tensor is actually not an equation but is an identity, called Jacobi identity and follows because of the antisymmetric nature of the tensor. It seems, thus the equation is a property of the antisym tensor itself and so probably not possible to derive from an action principle.