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A Four Function Variational Principle for Barotropic Magnetohydrodynamics

Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. In a previous work Yahalom & Lynden-Bell introduced a simpler Eulerian variational principles from which all the relevant equations of magnetohydrodynamics can be derived. The variational principle was given in terms of six independent functions for non-stationary flows and three independent functions for stationary flows. This is less then the seven variables which appear in the standard equations of magnetohydrodynamics which are the magnetic field $\vec B$ the velocity field $\vec v$ and the density $ρ$. In this work I will improve on the previous results showing that non-stationary magnetohydrodynamics should be described by four functions .