Ferguson. history of the calculus of variations.0402357.pdf

A Brief Survey of the History of the Calculus
of Variations and its Applications

James Ferguson
******@
University of Victoria

Abstract
In this paper, we trace the development of the theory of the calculus of variations.
From its roots in the work of Greek thinkers and continuing through to the
Renaissance, we see that advances in physics serve as a catalyst for developments
in the mathematical theory. From the 18th century onwards, the task of
establishing a rigourous framework of the calculus of variations is studied,
culminating in Hilbert’s work on the Dirichlet problem and the development of
optimal control theory. Finally, we make a brief tour of some applications of the
theory to diverse problems.


Introduction

Consider the following three problems:

1) What plane curve connecting two given points has the shortest length?

2) Given two points A and B in a vertical plane, find the path AMB which the
movable particle M will traverse in shortest time, assuming that its acceleration is
due only to gravity.

3) Find the minimum surface of revolution passing through two given fixed points,
(xA, yA) and (xB, yB).

All three of these problems can be solved by the calculus of variations. A field
developed primarily in the eighteenth and eenth centuries, the calculus of variations
has been applied to a myriad of physical and mathematical problems since its inception.
In a sense, it is a generalization of calculus. Essentially, the goal is to find a path, curve,
or surface for which a given function has a stationary value. In our three introductory
problems, for instance, this stationary value corresponds to a minimum.
The variety and diversity of the theory’s practical applications is quite astonishing. From
soap bubbles to the construction of an ideal column and from quantum field theory to
softer spacecraft landings, this venerable branch of mathematics has a