Elementary Characterization of Classical Minima

Abstract

many introductory treatments and is consequently ignored). It is therefore surprising that several optima of classical interest usually treated through the calculus of variations can be characterized directly by analysis much more elementary than the Euler-Lagrange theory. We shall substantiate this assertion by using little more than the Cauchy-Schwarz vector inequality to provide direct access to the solutions of several well-known problems, including certain of those associated with the brachistochrone and the minimal surface. We begin with two almost trivial examples from the classical literature [2, pp. 178-180, 201], namely those of determining geodesics in space and on the surface of a sphere. To find the shortest space curve joining fixed points P0(= 0) and PI, we must minimize I jP'(t)ldt