Newtonian non-linear hydrodynamics and magnetohydrodynamics

We use covariant techniques to study the non-linear evolution of self-gravitating, non-relativistic media. The formalism is first applied to imperfect fluids, aiming at the kinematic effects of viscosity, before extended to inhomogeneous magnetized environments. The non-linear electrodynamic formulae are derived and successively applied to electrically resistive and to highly conductive fluids. By nature, the covariant equations isolate the magnetic effects on the kinematics and the dynamics of the medium, combining mathematical transparency and physical clarity. Employing the Newtonian analogue of the relativistic 1 + 3 covariant treatment also facilitates the direct comparison with the earlier relativistic studies and helps to identify the differences in an unambiguous way. The purpose of this work is to set the framework and take a first step towards the detailed analytical study of complex non-linear systems, like non-relativistic astrophysical plasmas and collapsing protogalactic clouds. Key words: hydrodynamics – MHD. 1 IN T RO D U C T I O N General relativity is believed to describe strong gravitational field