Conservation Laws in Fluids and MHD: MultiSymplectic and Hamiltonian Approaches

Abstract:

We discuss conservation laws in ideal fluids and in magneto-hydrodynamics (MHD) using Eulerian, Lagrangian and Multi-Symplectic approaches. We show how the fluid and MHD equations can be written in multi-symplectic form using multi-momenta (the de Donder Weyl approach) in which both space and time are placed on an equal footing. We illustrate how this approach gives rise to both local and non-local conservation laws for both barotropic and non-barotropic gases. We obtain the symplecticity and pullback conservation laws for 1D gas dynamics and for the multi-dimensional gas and MHD equations. For the case of a non-barotropic gas, the helicity and cross helicity conservation laws are nonlocal conservation laws. A similar nonlocal conservation law also applies for 1D, non-barotropic gas dynamics, in which a nonlocal variable corresponding to the integral of the temperature back along the fluid path, keeps track of the history of the fluid element.