Living Forces and Stationary Action: Leibniz at the Limits of Analytic Mechanics

Although Leibniz’s reputation had something to do with the debate about the least action principle within the 18th century, he never stated it as such. He did however address the two conceptual elements in the principle. First, as an interpretation of divine providence in creation, he developed a mathematical theory of natural optimisation in his works on the optics, dynamics, and other physical subjects. Second, in his later dynamical works, he developed a theory of “action” (actio) which is the product of mass, displacement, and speed (a = msv). This second component is also situated within the context of vis viva conservation (mv2). With these two elements in place, and together with the conservation of vis viva, Leibniz seems to have arrived at the doorstep of the least action principle but never entered the open door. This paper argues that while Leibniz did not state a least action principle, his use of the concept of action together with the conservation vis viva implies stationary action, an eventual and more sophisticated interpretation of the least action principle. By reconstructing this argument, we can also show how Leibniz’s defence of a naturalistic, intrinsic teleology developed into a mathematical analysis of action quite distinct from the least action principles of the later Maupertuis and Euler. Instead of an optimisation formed by a sum over the path of motion, Leibniz’s method implies the actualisation of the optimum at each infinitesimal segment of the path. Finality, therefore, is not realised “at the end” but actualised continuously.