In this chapter we visit mechanics in the Age of Enlightenment. In that period, Newton's ideas, which allowed only the study of motion of bodies free in space, were extended to incorporate constraints in mechanical systems. The key figures are James Bernoulli, Jean le Rond d'Alembert and Joseph-Louis Lagrange. The central concept is the principle of virtual work, which establishes the conditions of static equilibrium and its extension to dynamics.

The Principle of Virtual Work

The idea of treating a static equilibrium problem using ideas from dynamics goes back to Aristotle's text “Mechanical Problems.” Although his authorship is disputed, it is probably the product of his contemporaries of the Peripatetic School. In the discussion of the lever – although the word equilibrium is never used – we read “the ratio of the weight moved to the weight moving it is the inverse ratio of the distances from the center” (Aristotle, 350 BC/1955, p. 353). This statement is regarded by many as a precursor to the so-called method of virtual velocities, or virtual displacements (Capecchi, 2012). In “On Mechanics,” one of his early works, (Galileo, 1600/1960) borrows Aristotle's idea and treats equilibrium on an inclined plane as an invariance under hypothetical displacements. In the “Discourses,” he uses notions of statics and dynamics in the same sentence: “when equilibrium (that is, rest) is to prevail between two moveables, their [overall] speeds or their propensities to motion – that is, the spaces they would pass in the same time – must be inverse to their weights [gravità]”(Galilei, 1638/1974, p. 173). Since Galileo talks about the “propensity” to move, and the system is at rest, the velocity refers to a hypothetical motion in a time different from the time of our universe. Galileo realized that, for weights on an inclined plane, the determining factor for equilibrium is their motion away from or “removal from the center of the earth”(Galileo, 1600/1960, p. 177). For the inclined plane with two masses of Figure 5.1, the ratio of the masses is 2. In order for the system to be at equilibrium, the ratio of the vertical velocities (if they were displaced in the same amount of time) is 1/2.