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 ${\mathcal{L}}_{m}=\ensuremath{-}\ensuremath{\rho}[{c}^{2}+\ensuremath{\int}P(\ensuremath{\rho})/{\ensuremath{\rho}}^{2}d\ensuremath{\rho}]$, where $\ensuremath{\rho}$ is the rest mass density and $P(\ensuremath{\rho})$ 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 (${\ensuremath{\nabla}}_{\ensuremath{\sigma}}(\ensuremath{\rho}{u}^{\ensuremath{\sigma}})=0$).