History of Virtual Work Laws by Danilo Capecchi

Author:Danilo Capecchi
Language: eng
Format: epub
Publisher: Springer Milan, Milano

(10.14)

that he names general equation of equilibrium. Note that in this equation dp, dq, dr etc. can vary freely, as if there were no constraints attached.

In the next sections of the Mécanique, Lagrange addresses problems of some interest in mechanics, including Maupertuis’s law of rest; they do not cover basic aspects of virtual work laws and therefore will be ignored.

10.2.1.1 Constraint reactions

In mechanics, the idea of constraint reactions has evolved along with those of force and constraint. For Aristotle, a constraint was essentially an impediment for a body to reach its natural place. Removing the constraint leaves the body free to move. Even among the ancient Greeks, especially among engineers, it was clear that the effect of a constraint could be obtained with a power, a muscular force for instance. If a heavy body was fixed on a hook by means of a rope, it was clear that the role of the hook could be played by a muscular force, appropriate to support the weight. Therefore, the possibility of interchangeability between constraints and powers is seen from the beginning of mechanics, although they remain distinct concepts, the constraint does not exert a force on a body but it seems as if it does. One can talk about constraint reaction as the force that, for equilibrium, has the same effect of the constraint.

By accepting the rule of the parallelogram as the primary tool for addressing the study of static problems, which occurred due to Varignon’s Nouvelle mécanique ou statique of 1725, the reaction forces in this sense began to appear explicitly as geometric or algebraic variables in the calculations. As the equilibrium reduces to the annulment of the sum of the forces, if also a constraint contributes to the equilibrium, there is nothing more natural than in the equations of equilibrium symbols appearing to represent the forces equivalent to constraints, e.g. the reactive forces. Only after Newton, with the introduction of forces at a distance as physical magnitudes, the principle of action and reaction and the emergence of the corpuscular concept of matter and the theory of elasticity, did the ontological status of the reactive forces begin to change. The constraints are no longer, in general, impediments to motion but they become bodies composed of particles that are centres of forces and the constraint reactions are ‘real’ forces that the constraint-body exert over other bodies which are to interact with them.

In the XVIII century, constraints are still generally modeled as hard bodies, that is, as being capable of absorbing motions and impulses acting at right angles to them, but toward the end of the century with the emergence of the Eulerian and Newtonian concepts of force, constraints are entities treated as dispensers of forces and thus no longer merely passive. By the XIX century, this second point of view becomes prevalent, especially among the French scientists.

Lagrange is situated in an intermediate position, on the one hand he considers constraint forces as the forces required to perform the functions of the constraints, on the other hand he gives them the ontological status of active forces, or powers.

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