Variational Formulations for Non-Relativistic and Relativistic Fluid Motions in an Eulerian Context

In its first part, this article develops a variational formulation for a fluid motion in a nonrelativistic context. The results are obtained through standard tools of calculus of variations and constrained optimization in function spaces. In order to include a scalar field of temperature as a variable, concerning the main functional, we assume a rather standard energy equation as a constraint. In the last section, we also address a fluid motion in a relativistic context. Finally, we emphasize the modeling approach here addressed is essentially an Eulerian one.