We apply Noethers first and second theorems to investigate conservation laws in magnetohydrodynamics (MHD) and gas dynamics. A version of Noethers second theoremusing Lagrange multipliers is used to investigate fluid relabelling symmetries conservation laws. Ertels 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 Noethers 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 Ertels theorem in ideal fluid mechanics. An independent derivation shows that the new conservation law is also valid for more general physical situations. #MindGarden #Phasespace #OrganicComputersInHyperspace #ComplexityScience
Posted on: Fri, 14 Mar 2014 10:42:06 +0000
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