Noether's Theorems and Fluid Relabelling Symmetries in Magnetohydrodynamics and Gas Dynamics

We apply Noether's first and second theorems to investigate conservation laws in magnetohydrodynamics (MHD) and gas dynamics. A version of Noether's second theorem using Lagrange multipliers is used to investigate fluid relabelling symmetries conservation laws. Ertel's theorem, the fluid helicity conservation equation and cross helicity conservation equation for a barotropic gas are obtained using Lagrange multipliers which are used to enforce the fluid relabelling symmetry determining equations. The nonlocal form of the non-magnetized fluid helicity conservation law and the nonlocal cross helicity conservation laws obtained previously are briefly discussed. We obtain a new generalized potential vorticity type conservation equation for MHD which takes into account entropy gradients and the J ×B force on the plasma due to the current J and magnetic induction B. This new conservation law for MHD is derived by using Noether's second theorem in conjunction with a class of fluid relabelling symmetries in which the symmetry generator for the Lagrange label transformations is non-parallel to the magnetic field induction in Lagrange label space. This is associated with an Abelian Lie pseudo algebra and a foliated phase space in Lagrange label space. It contains as a special case Ertel's theorem in ideal fluid mechanics. An independent derivation shows that the new conservation law is also valid for more general physical situations.