Variational principle for the relativistic hydrodynamic flows with discontinuities, and local invariants of motion

A rigorous method for introducing the variational principle describing relativistic ideal hydrodynamic flows with all possible types of discontinuities (including shocks) is presented in the framework of an exact Clebsch type representation of the four-velocity field as a bilinear combination of the scalar fields. The boundary conditions for these fields on the discontinuities are found. We also discuss the local invariants caused by the relabeling symmetry of the problem and derive recursion relations linking invariants of different types. These invariants are of specific interest for stability problems. In particular, we present a set of invariants based on the relativistic generalization of the Ertel invariant.