Variational principle of relativistic perfect fluid
  We reformulate the relativistic perfect fluid system on curved space-time. Using standard variables, the velocity field $u$,energy density $\rho$ and pressure $p$, the covariant Euler-Lagrange equation is obtained from variational principle. This leads to the Euler equation and the equation of continuity in reparametrization invariant form. 
Takayoshi Ootsuka, Muneyuki Ishida, Erico Tanaka, Ryoko Yahagi
gr-qc math-ph math.MP physics.flu-dyn