A partial report on the controversies about the Principle of Virtual Work: from Archytas of Tarentum to Lagrange, Piola, Mindlin and Toupin

In this chapter, we present a historical survey on the Principle of Virtual Work as the guiding principle for constructing mathematical models to describe and predict phenomena. In particular, we want to make the reader aware of the development of two main approaches toward the formulation of new models, the first using the Principle of Virtual Work, while the second using the balance of some quantities to be suitably chosen. This dualism, which was probably already present in Hellenistic times, has been nurtured in modern times by some of the most important scientists of the last centuries. We think it is worthwhile to study their work, not only to attempt a historically-founded authorship attribution of the underlying ideas, but especially so that their work can guide us while developing new theories. Our discussion, which begins from the available fragmentary sources dealing with Hellenistic Mechanics, focuses on the efforts by D'Alembert and Lagrange, which produced a modern comprehensive formulation of Classical Mechanics based on the Principle of Virtual Work. Referring to these historical instances, we advocate the effectiveness of using the Principle of Virtual Work, as opposed to balance laws, as a basic postulate in formulating new models, arguing that it allows using a minimal set of a priori and clear conjectures, avoiding the need for a posteriori ad hoc - hence often logically incompatible - assumptions based on "physical intuition". As a paradigmatic case, we present the formulation of N-th Gradient Continuum Mechanics developed in the pioneering - but not yet widely known - work of Piola, where a wise use of the Principle of Virtual Work leads to a theory that is more general than the one developed starting from Cauchy’s "tetrahedron argument", based on the Law of Balance.