Leibniz and the Calculus of Variations

Abstract

The brachistochrone problem of Johann Bernoulli is considered as the origin of the calculus of variations. The solutions presented by Johann and Jacob Bernoulli and by Newton and Leibniz were all different and highly original. Leibniz’ solution has received less attention than those of the Bernoullis, but I show here that his abstract idea was also general and powerful enough for a general theory, although the history of mathematics took a different path. In fact, his approach quite naturally emerges from his earlier treatment of the refraction of light by his then new calculus, i.e., his derivation of Fermat’s principle. I then analyze the development of his conceptions about the speed of light from that treatment through his work on the brachistochrone problem to his Nouveaux essais from 1706. From the work of the Bernoullis and Leibniz on variational problems, also an analogy between mechanical and optical problems emerged, and this naturally leads to Leibniz’ considerations on the physical concept of action and extremal principles. In contrast to later formulations of such a principle by Maupertuis and Euler, Leibniz devoted much effort to deriving more abstract principles based on considerations of symmetry and determination, as analyzed in De Risi (Geometry and monadology: Leibniz’s analysis situs and philosophy of space. Birkhäuser, Basel/Boston, 2007). Some of his corresponding ideas look surprisingly modern, for instance in the light of Feynman’s path integral approach to quantum mechanics. Leibniz’ ideas are put into the perspective of modern science in Jost (Leibniz und die moderne Naturwissenschaft. Springer, 2019).