New derivation of the Lagrangian of a perfect fluid with a barotropic equation of state

In this paper we give a simple proof that when the particle number is conserved, the Lagrangian of a barotropic perfect fluid is ${\mathcal{L}}_m=-\rho [c^2+\int P(\rho )/{\rho }^{2}d\rho ]$, where $\rho$ is the rest mass density and $P(\rho )$ is the pressure. To prove this result, neither additional fields nor Lagrange multipliers are needed. Besides, the result is applicable to a wide range of theories of gravitation. The only assumptions used in the derivation are: 1) the matter part of the Lagrangian does not depend on the derivatives of the metric, and 2) the particle number of the fluid is conserved (${\nabla }_{\sigma }(\rho u^{\sigma })=0$).