  Variational principles for magnetohydrodynamics (MHD) were introduced by
previous authors both in Lagrangian and Eulerian form. In this paper we
introduce simpler Eulerian variational principles from which all the relevant
equations of non-barotropic MHD can be derived for certain field topologies.
  The variational principle is given in terms of five independent functions for
non-stationary non-barotropic flows. This is less then the eight variables
which appear in the standard equations of barotropic MHD which are the magnetic
field $\vec B$ the velocity field $\vec v$, the entropy $s$ and the density
$\rho$.
  The case of non-barotropic MHD in which the internal energy is a function of
both entropy and density was not discussed in previous works which were
concerned with the simplistic barotropic case. It is important to understand
the rule of entropy and temperature for the variational analysis of MHD. Thus
we introduce a variational principle of non-barotropic MHD and show that five
functions will suffice to describe this physical system.
  We will also discuss the implications of the above analysis for topological
constants. It will be shown that while cross helicity is not conserved for
non-barotropic MHD a variant of this quantity is. The implications of this to
non-barotropic MHD stability is discussed.
