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

Variational principle for the relativistic hydrodynamic flows with discontinuities, and local invariants of motion. Abstract 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. Introduction. In this paper we discuss some problems related to ideal relativistic hydro-dynamic (RHD) flows in the framework of the special relativity. They are pertinent to the description of flows with discontinuities, including shocks, in terms of canonical (Hamiltonian) variables based upon the corresponding variational principle and introducing local invariants along with recursion relations. These subjects are of interest from a general point of view and are very useful in solving nonlinear problems, specifically, nonlinear stability investigation, description of the turbulent flows, etc. In particular, the use of the Hamiltonian approach along with additional local invariants of the motion and the corresponding Casimirs allows to improve the nonlinear stability criteria. The necessity to consider the relativistic flows is motivated by a wide area of applications, including the astrophysical and cosmological problems.