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 \textit{rest mass}
density and $P(\rho)$ is the pressure. To prove this result nor additional
fields neither 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 ($\nabla_\sigma (\rho u^\sigma)=0$).
