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.

 

[astrokreng]

The derivation of the Maxwell equations in the source-free case is a fundamental aspect of electromagnetic theory. These equations describe the behavior of electric and magnetic fields in the absence of any sources, such as charges or currents. The first equation, \partial_{\mu}F^{\mu \nu}=0, can be derived from the action principle by varying the action with respect to the vector potential A_{\mu}. This yields the familiar expression for the electromagnetic field tensor, F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}. However, the second equation, \partial_{\mu}\tilde{F}^{\mu \nu}=0, is not as straightforward to derive.

To understand the derivation of the second equation, we need to first understand the significance of the dual field tensor, \tilde{F}^{\mu \nu}. This tensor is related to the original field tensor, F^{\mu \nu}, through the duality transformation, \tilde{F}^{\mu \nu}=\frac{1}{2}\epsilon^{\mu \nu \alpha \beta}F_{\alpha \beta}, where \epsilon^{\mu \nu \alpha \beta} is the Levi-Civita symbol. This transformation is an important mathematical tool in electromagnetism, as it allows us to express the equations of electromagnetism in a more symmetric and elegant way.

To derive the second equation, we need to consider the action principle in terms of the dual field tensor. This can be done by first expressing the action in terms of the electric and magnetic fields, E and B, and then using the duality transformation to rewrite it in terms of the dual field tensor. By varying this action with respect to the dual field tensor, we can obtain the second Maxwell equation.

It is important to note that both the original field tensor and the dual field tensor are fundamental quantities in electromagnetism. They are related to each other through the duality transformation and cannot be considered independent of each other. Both tensors play a crucial role in describing the behavior of electromagnetic fields and cannot be disregarded.

In summary, the second Maxwell equation, \partial_{\mu}\tilde{F}^{\mu \nu}=0, can be derived from the action principle by considering the dual field tensor and using the duality transformation. Both the original field tensor and