This article develops a variational formulation for a fluid motion in a non-relativistic 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. Finally, we emphasize the modeling context here addressed is essentially an Eulerian one.

In its first part, this article develops a variational formulation for the incompressible Euler system in fluid mechanics. The results are based on standard tools of calculus of variations and constrained optimization. In a second step, we present a variational formulation for a compressible Euler system in fluid mechanics assuming an approximately constant scalar field of temperature. In the subsequent sections we also present variational formulations for the Navier-Stokes system and for a relativistic fluid motion. Finally, in the last sections, we develop a duality principle applied to a Ginzburg-Landau type equation.

In its first part, this article develops a variational formulation for the incompressible Euler system in fluid mechanics. The results are based on standard tools of calculus of variations and constrained optimization. In a second step, we present a variational formulation for a compressible Euler system in fluid mechanics assuming an approximately constant scalar field of temperature. In the subsequent sections we also present a variational formulation for a relativistic fluid motion. Finally, in the last sections, we develop a duality principle applied to a Ginzburg-Landau type equation.