Calculus of Variations

The calculus of variations is a branch of mathematical analysis that studies extrema and critical points of functionals (or energies). Here, by functional we mean a mapping from a function space to the real numbers. One of the first questions that may be framed within this theory is Dido’s isoperimetric problem (see Subsection 2.3): to find the shape of a curve of prescribed perimeter that maximizes the area enclosed. Dido was a Phoenician princess who emigrated to North Africa and upon arrival obtained from the native chief as much territory as she could enclose with an ox hide. She cut the hide into a long strip, and used it to delineate the territory later known as Carthage, bounded by a straight coastal line and a semi-circle. It is commonly accepted that the systematic development of the theory of the calculus of variations began with the brachistochrone curve problem proposed by Johann Bernoulli in 1696: consider two points A and B on the same vertical plane but on different vertical lines. Assume that A is higher than B, and that a particle M is moving from A to B along a curve and under the action of gravity. The curve that minimizes the time travelled by M is called the brachistochrone. The solution to this problem required the use of infinitesimal calculus and was later found by Jacob Bernoulli, Newton, Leibniz and de l’Hôpital. The arguments thus developed led to the development of the foundations of the calculus of variations by Euler. Important contributions to the subject are attributed to Dirichlet, Hilbert, Lebesgue, Riemann, Tonelli, Weierstrass, among many others. The common feature underlying Dido’s and the brachistochrone problems is that one seeks to maximize or minimize a functional over a class of competitors satisfying given constraints. In both cases the functional is given by an integral of a density depending on an underlying field and some of its derivatives, and this will be the prototype we will adopt in what follows. Precisely, we consider a functional