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In this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time. We show that a relaxed version of this variational problem always has a solution, and we derive an Euler-Lagrange system for the relaxed minimization problem which we call the relaxed Euler equations. Finally, we prove consistency between the relaxed Euler equations and the classical Euler system, showing that weak solutions of the relaxed Euler equations with the appropriate geometric structure give rise to classical Euler solutions and that classical solutions of the Euler system induce weak solutions of the relaxed Euler equations. The first consistency result is new even in the constant density case. The remainder of our analysis is an extension of the work of Y. Brenier (1999) to the variable density case

Least action principle and the incompressible Euler equations with variable density