Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether’s theorem. The text includes numerous examples along with problems to help students consolidate the material.

can always be made arbitrarily small when convenient. The auxiliary set H can thus be replaced by the set H = {η ∈ X : y + η ∈ S}, for the purposes of analysis. At this stage we specialize to a particular class of problem called the ﬁxed endpoint variational problem, 3 and work with the vector space C 2 [x0 , x1 ] that consists of functions on [x0 , x1 ] that have continuous second derivatives. Let J : C 2 [x0 , x1 ] → R be a functional of the form 2 3 See Appendix B.1. More accurately, it is

const., y0 = const., and x1 = const. These equations are then supplemented by the natural boundary condition at (x1 , y1 ) to provide the fourth equation. In this problem only the variation δy at (x1 , y1 ) is arbitrary. Typically, variational problems come with relations of the form gk (xj , yj ) = 0, (7.24) for j = 1, 2 so that the endpoint variations of (x0 , y0 ) are not linked to those of (x1 , y1 ). In this case we can always include variations that leave one endpoint ﬁxed, and this leads

the cable assumes when supported between the poles. The problem was posed by Jacob Bernoulli in 1690. By the end of 1691 the problem was solved by Leibniz, Huygens, and Jacob’s younger brother Johann Bernoulli. It should be noted that Galileo had earlier considered the problem, but he thought the catenary was essentially a parabola.4 Since the arclength L of the cable is given, we can use expression (1.1) to look for a minimum potential energy conﬁguration. Instead, we start with expression

Principle he writes, “It appears that Maupertuis reached this obscure expression by an unclear mingling of his ideas of vis viva and the principle of virtual velocities” (p. 365). In defense of Mach, we must note that Maupertuis suﬀered no lack of critics even in his own day. Voltaire wrote the satire Histoire du docteur Akakia et du naif de Saint Malo about Maupertuis. The situation at Frederick the Great’s court regarding Maupertuis, K¨ onig, and Voltaire is the stuﬀ of soap operas (see Pars

the existence of such a ﬁgure. Weierstraß and his followers resolved these subtle aspects of the problem. A lively account of Dido’s problem and the ﬁrst of Steiner’s proofs can be found in K¨ orner [45]. Some simple geometrical arguments can be used to show that if γ is a simple closed curve solution to Dido’s problem then γ is convex (cf. K¨orner, op. cit.). This means that a chord joining any two points on γ lies within γ 10 11 The reader will ﬁnd various bits and pieces of Dido’s history