For simplicity, let's assume the Lagrangian formulation of Noether's theorem, that is, our equations of motion can be derived from the Euler-Lagrange equations, or, simply, that we can use a Lagrangian to properly describe the motion of our system.
Briefly, the theorem states that a continuous symmetry of the action of this system gives a conserved quantity. Or, to every continuous symmetry of the action there's a conservation law attached.
Well, let's take a look at the Poynting Vector in the vacuum:
\begin{equation}
\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}
\end{equation}
This vector gives the direction of propagation of the electromagnetic energy in the vacuum. Going further, we do state a conservation law for the energy in a electromagnetic field, that's Poynting's Theorem:
\begin{equation}
\nabla\cdot\mathbf{S}\ +\ \mathbf{J}\cdot\mathbf{E}=-\dfrac{\partial u}{\partial t}
\end{equation}
What I want know is, under what transformation is the Lagrangian of such a system symmetric, that makes this conservation law arise?
I'm not particularly looking for a demonstration of the diff equation which represents the conservation law, I'm more interested in understanding the symmetry here.
Best Answer
As it is the case in Classical Mechanics, the conservation of energy in Electromagnetism (which is expressed by means of Poynting's Theorem) is a consequence of time-translation invariance. When working with the Noether Theorem in Field Theory, it is often more interesting to do things covariantly and prove conservation of another quantity, the symmetric stress-energy tensor $T^{\mu\nu}$. The energy density will then be given by $T^{00}$, the Poynting vector is associated with $T^{0i} = T^{i0}$ and the remaining components, $T^{ij}$, are associated with Maxwell's stress tensor.
If I am not mistaken, this approach to the conservation laws of E&M is discussed, e.g., on K. Lechner's Classical Electrodynamics: A Modern Perspective.