CHAPTER 2THE CALCULUS OF VARIATIONS

CHAPTER 2
THE CALCULUS OF VARIATIONS
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. hav:fng been
promised all the land lying within the boundaries of a hull's hide. the
clever queen cut the hide into many thin strips. tied them together in
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such a way as to secure as much land as possible within this boundary.
The solution is of course a circle. This is a typical isoperimetric
probleQ 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. dl! 1 '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
In Economics, the use of the Calculus of Variations goes back
to the 1920's with the works of Evans (1924, 1930), Ramsey (1928) and
Hotelling (1931). Evans and Roos attempted to find the optimal price
path for the whole planning period such as to maximize the profit
functional of the monopolist. This is a typical problem of the Calculus of