Relativistic least action principle for discontinuous hydrodynamic flows, Hamiltonian variables, helicity and Ertel invariant

A rigorous method for introducing the variational principle describing relativistic ideal hydrodynamic flows with all possible types of breaks (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 breaks are found. We also discuss the local invariants caused by the symmetries of the problem, including relabeling symmetry. In particular, the generalization of the well-known nonrelativistic Ertel invariant is presented.