CONSTRUCTING THE LAGRANGIAN IN THE EULERIAN COORDINATE FOR RELATIVISTIC HYDRODYNAMICS

The Lorentz-covariant Lagrangian for an ideal relativistic flow is constructed in the Eulerian coordinate. In contrast to the Lagrangian of nonrelativistic flows in the Eulerian formulation, for which the continuity equation is required to be externally imposed as a constraint [Mittag, Stephen, and Yourgrau, in Variational Principles in Dynamics and Quantum Theory, edited by W. Yourgrau and X. Mandelstam (Dover, New York, 1968)), this Lorentz-covariant Lagrangian automatically yields the continuity equation as well as the equation of state. In addition, the relativistic generalization of the Bernoulli equation can also be derived from the present formulation.