# Which symmetry is associated with conservation of flux?

Which symmetry is associated with conservation of flux (e.g., in electromagnetism)?

For example, when working with Gauss's law in electromagnetism, net flux through an arbitrary volume element remains unchanged when the net charge is unchanged.

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Could you give a different example? The example you gave is not a conservation law, but merely an integral formulation of the fact that $\nabla^2 \phi=\rho$. Also, when introducing relativistic effects, it is not true anymore. – yohBS Jan 4 '12 at 20:36
@yohBS: actually, the Gauss constraint continues to hold if you remember that the differential version is the infinitesimal version of $\oint_S E \cdot dS = Q$, and you boost $E$ and $S$ appropriately, and use the charge $Q$ in the rest frame of $S$. – genneth Jan 4 '12 at 22:05
$divE=\rho$ does not have relativistic corrections. If the charge is localized in some volume, the integral form of this equation gives the total charge that does not change. – Vladimir Kalitvianski Jan 4 '12 at 23:23

Which symmetry is associated with conservation of electric and magnetic fluxes $\Phi_E$ and $\Phi_B$, respectively?
Fluxes $\Phi_E$ and $\Phi_B$ are integral quantities. The corresponding differential quantities are the two first Maxwell equations $\vec{\nabla}\cdot\vec{E}-\rho=0$ and $\vec{\nabla}\cdot\vec{B}=0$, respectively. The former is a first class constraint that generates gauge symmetry, while there is no symmetry associated with the latter, as it is a Bianchi identity.
Qmechanic gives the Hamiltonian formalism. In the Lagrangian formalism, we get the Gauss law $\nabla \cdot E = \rho$ by applying Noether's 2nd theorem, which is often neglected in textbooks, to the gauge symmetry in electrodynamics. Importantly, the 2nd theorem gives "conservation" laws which are independent of the equations of motion, and are sometimes referred to as "off-shell". From the point of view of the Hamiltonian formalism, we see that if we wanted to specify the initial data from which time evolution of the system is entirely determined, we must satisfy the constraint for the problem to be well-defined.