Variational Methods for Engineers with Matlab

7Functionals and Calculus of Variations

For people interested in ancient myths, calculus of variations may be dated from the creation of Carthage by Elissa (also called Dido). This mythical phenycian queen which “bought all the land she could enclosed with a bull’s hide” (“mercatique solum, facti de nomine Byrsam, taurine quantum possent circumdare tergo” [VIR 19]), delimited a circle using a string made of tires cut in a bull’s hide. In fact, philosophers have long considered the connections between area and perimeter and looked for the maximal area for a given length (namely Zenodorus the Greek and Pappus of Alexandria) and Hero of Alexandria formulated a principle of economy, which corresponds to a variational principle.

Other people date calculus of variations from the famous problem stated by Johann Bernouilli about the brachistochrone ([BER 96], even if the same problem had already been proposed by Galilei Galileo precedently [GAL 38, ERL 88] – in addition, Fermat and Huyghens previously formulated physical principles leading to problems of Calculus of variations.

The basic problem of calculus of variations consists of determining a field that realizes the minimum or the maximum of a real quantity associated with it, such as, for instance, the energy, the area, the length, a characteristic time, etc. In the case of a real function f: → , such a point x verifies f'(x) = 0, ...