Least Action and Virtual Work Principles for the Formulation of Generalized Continuum Models

Although the principle of virtual velocities is rooted deep in time, the study of its historical development, as well as studies of its variations and transformations that have been presented at various times, are surprisingly meagre. In this rather sparingly explored field we have found especially interesting the work [36], and particularly modern and still topical the presentations by Lagrange and Piola ([15, 30],[24, 25, 27–29]). A presentation of the history of mechanics cataloguing the main contributions to mechanical sciences on the basis of their postulation scheme is, in our opinion, a worthy endeavor that deserves the exemplary efforts of modern mechanicians. It is clear that the great majority of formulations of novel branches of mechanics, and more generally of physics, has been obtained via a basic unifying variational principle (see eg [1, 11, 16, 18, 26, 32–35]). In his works Jean Le Rond D’Alembert has been one of the bravest champions of this point of view: we refer to [10] for a due tribute to his contribution to mechanical sciences. The D’Alembertian point of view is shared, among many other scholars, by Hellinger ([38]), Landau ([17]), Feynman [11], Germain [13], Cosserat brothers [6], Lagrange [30], Piola [29], Sedov, Mindlin and Toupin [28], and Winter [23].