Hamilton-jacobi equation for relativistic charged fluid flow

Abstract

Representing the viscous and heat-injection 4-forces acting on a charged fluid in terms of the thermal 4-potential introduced in the preceding paper makes possible the derivation of a scalar equation of motion for the fluid that has the form of a generalization of the relativistic Hamilton-Jacobi equation for a charged particle in the presence of gravitational and electromagnetic fields. In this generalized Hamilton-Jacobi equation the specific enthalpy and the thermal 4-potential play roles that are analogous to the scalar gravitational potential and to the electromagnetic 4-potential respectively. The formalism employs a generalization of the canonical particle momentum that includes the thermal 4-potential as well as the electromagnetic 4-potential. This generalized canonical momentum can be represented in terms of three scalar functions by means of the Clebsch transformation. One of these scalar functions is Hamilton’s characteristic function, which is the unknown in the generalized Hamilton-Jacobi equation. The other two functions, which are called the vorticity invariants, determine the intrinsic vorticity of the fluid, which is defined as the curl of the generalized canonical momentum and which, according to the generalized Larmor theorem derived in the preceding paper, is to be associated with that part of the fluid rotation that is a residual of the initial conditions of the fluid. The vorticity invariants are both constants of motion of the fluid. In the case of adiabatic flow, it is possible to express the thermal 4-potential in terms of the specific entropy and the temperature integral, which is defined as the scalar function whose substantial time derivative is equal to the temperature. This allows a simple interpretation of the generalized canonical momentum in terms of the moving heat reservoir model introduced in a previous paper.