Least action principle for . . .

As Piola would have surely conjectured, the stationary action principle holds also for capillary fluids, i.e. those fluids for which the deformation energy depends on spatial derivative of mass density (a modelling necessity which has been already remarked by Cahn and Hilliard [15, 16]). For capillary fluids it is indeed possible to define a Lagrangian density function whose corresponding Euler-Lagrange stationarity conditions once transported on the actual configuration, via a Piola's transformation, are exactly those obtained, with different methods, in the literature. We recall that some particulat classes of second gradient fluids are sometimes also called Korteweg-de Vries or Cahn-Allen fluids. More generally those continua (which may be solid or fluid) whose deformation energy depends on the secon