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 Lₘ=-ρ[c²+∫P(ρ)/ρ²dρ], where ρ is the rest mass density and P(ρ) 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 (∇_σ(ρu^σ)=0).