Lecture 1 The Brachistochrone Although the roots of the calculus of variations can be traced to much earlier times, the birth date of the subject is widely considered to be June of 1696.1 That is when John Bernoulli posed the celebrated problem of the brachistochrone or curve of quickest descent, i.e., to determine the shape of a smooth wire on which a frictionless bead slides between two fixed points in the shortest possible time. x 0 0.5 1 y 0 0.5 1 Figure 1.1. A frictionless bead on a wire. For the sake of definiteness, let us suppose that the points in question have coordinates (0, 1) and (1, 0), and that the bead slides 1 See, e.g., Bliss [5, pp. 12-13 and 174-179] or Hildebrandt & Tromba [21, pp. 26- 27 and 120-123], although Goldstine [17, p. vii] prefers the earlier date of 1662 when Fermat applied his principle of least time to light ray refraction. 1 http://dx.doi.org/10.1090/stml/050/01

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2009 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.