A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense
According to Eneström, Euler completed the manuscript of this work by April 1743.
This work is concerned with the calculus of variations. Euler's main contribution to this subject was to change it from a discussion of essentially special cases to a discussion of very general classes of problems. This work includes a listing of 100 special problems that illustrate his methods. Euler also demonstrates a general procedure for writing down the so-called "Euler differential equation" or first necessary condition. Some problems that Euler uses to demonstrate his methods are: (1) find, among all plane curves y = y(x), 0 ≤ x ≤ a, the one that maximizes or minimizes ∫Z dx, where Z is a "determinate" function of x, y, and y's derivatives; (2) find the shape of brachystochrone and tautochrone curves in a resistant medium; (3) find the geodesic joining two fixed points on a given concave or convex surface. In solving the brachystochrone problem, Euler finds a simple instance (possibly the first) of the Lagrange multiplier method. This is also the first work in which the principle of least action is presented, which Euler expresses in the following way. For a given projected body, denote its mass by M, half the square of its velocity by v, the arclength element by ds. Then, among all curves passing through the same pair of endpoints, the desired curve is the one that minimizes the integral ∫Mv1/2 ds. For a more detailed explanation, see Herman Goldstine's A History of the Calculus of Variations from the 17th through the 19th Century (Springer-Verlag, 1980).
Original Source Citation
Lausanne & Geneva: Marcum-Michaelem Bousquet, Volume 1744, pp. 1-322.
Opera Omnia Citation
Series 1, Volume 24, pp.1-308.