Standard covariant formulation for perfect-fluid dynamics

Abstract

After a brief description of the Milne generalization of the Galilean invariance group for the space–time of Newtonian kinematics, it is shown how the generalized Eulerian dynamical equations for the motion of a multiconstituent perfect (nonconducting) fluid can be expressed in terms of interior products of current 4-vectors with exterior derivatives of the appropriate 4-momentum 1-forms (whose role is central in this approach) in a fully covariant standard form whose structure is identical in the Newtonian case to that of the corresponding equation for the case of (special or general) relativistic perfect fluid mechanics. In addition to space–time covariance, this standard form exhibits chemical covariance in the sense that it is manifestly invariant under redefinition of the number densities of the independent conserved chemical constituents in terms of linear combinations of each other. It is shown how, in the strictly conservative case when no chemical reactions occur, this standard form, can be used (via the construction of suitably generalized Clebsch potentials) for setting up an Eulerian (fixed-point) variation principle in a form that is simultaneously valid for both Newtonian and relativistic cases.