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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.

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  • $\begingroup$ You even say it's energy conservation - what's special about Poynting's theorem that you think it's not just time translation symmetry like any other energy conservation law? $\endgroup$
    – ACuriousMind
    Commented Nov 26, 2021 at 22:12
  • $\begingroup$ Related: physics.stackexchange.com/q/454140/2451 $\endgroup$
    – Qmechanic
    Commented Nov 26, 2021 at 22:12

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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.

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