The Calculus of Variations is a branch of Mathematics dealing with optimization of functionals. The variational problem goes back to the antiquity. The first solution seems to have been that of queen Dido of Carthage in about 850 B.C. Virgil reported that, having been promised all the land lying within the boundaries of a bull’s hide, the clever queen cut the hide into many thin strips, tied them together in such a way as to secure as much land as possible within this boundary.1 The solution is of course a circle. This is a typical isoperimetric problem of the Calculus of Variations. However, it was not until the late seventeenth century that substantial progress was made when a rigorous solution of the brachistochrone problem was provided by Newton, de l’Hospital, John and Jacob Bernouilli in 1696. This problem consists of determining the shape of a curve joining A to B such that a frictionless particle sliding along it under the influence of gravity alone moves from A to B in the shortest time. The solution is a cycloid. This played an important part in the development of the Calculus of Variations.2
KeywordsEuler Equation Isoperimetric Problem Fundamental Lemma Satisfy Boundary Condition Point Constraint
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