# Noether's Theorem and Poynting Theorem

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: $$$$\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$$$$ This vector gives the direction of propagation of the electromagnetic energy in the vacuum. Going further, we do state a conservation law for the energy in a electromagnetic field, that's Poynting's Theorem: $$$$\nabla\cdot\mathbf{S}\ +\ \mathbf{J}\cdot\mathbf{E}=-\dfrac{\partial u}{\partial t}$$$$ 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.

• 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? Commented Nov 26, 2021 at 22:12
• Commented Nov 26, 2021 at 22:12

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.