Page 14 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 0

Introduction

0.1 Brief historical comments

The calculus of variations is one of the classical branches of mathematics. It was
Euler who, looking at the work of Lagrange, gave the present name, not really
self explanatory, to this field of mathematics.
In fact the subject is much older. It starts with one of the oldest problems in
mathematics: the isoperimetric inequality. A variant of this inequality is known
as the Dido problem (Dido was a semi historical Phoenician princess and later
a Carthaginian queen). Several more or less rigorous proofs were known since
the times of Zenodorus around 200 BC, who proved the inequality for polygons.
There are also significant contributions by Archimedes and Pappus. Impor-
tant attempts for proving the inequality are due to Euler, Galileo, Legendre,
L’Huilier, Riccati, Simpson or Steiner. The first proof that agrees with modern
standards is due to Weierstrass and it has been extended or proved with dif-
ferent tools by Blaschke, Bonnesen, Carathéodory, Edler, Frobenius, Hurwitz,
Lebesgue, Liebmann, Minkowski, H.A. Schwarz, Sturm, and Tonelli among oth-
ers. We refer to Porter [86] for an interesting article on the history of the
inequality.
Other important problems of the calculus of variations were considered in
the seventeenth century in Europe, such as the work of Fermat on geometrical
optics (1662), the problem of Newton (1685) for the study of bodies moving
in fluids (see also Huygens in 1691 on the same problem) or the problem of
the brachistochrone formulated by Galileo in 1638. This last problem had a
very strong influence on the development of the calculus of variations. It was
resolved by John Bernoulli in 1696 and almost immediately after also by James,
his brother, Leibniz and Newton. A decisive step was achieved with the work of

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