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

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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)/ρ² dρ], where ρ is the rest mass density and P(ρ) 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 (∇σ(ρuσ)=0).