This monograph explores the early development of the calculus of
variations in continental Europe during the Eighteenth Century by
illustrating the mathematics of its founders. Closely following the
original papers and correspondences of Euler, Lagrange, the Bernoullis,
and others, the reader is immersed in the challenge of theory building.
We see what the founders were doing, the difficulties they faced, the
mistakes they made, and their triumphs. The authors guide the reader
through these works with instructive commentaries and complements to the
original proofs, as well as offering a modern perspective where useful.
The authors begin in 1697 with Johann Bernoulli's work on the
brachystochrone problem and the events leading up to it, marking the
dawn of the calculus of variations. From there, they cover key advances
in the theory up to the development of Lagrange's δ-calculus, including:
- The isoperimetrical problems
- Shortest lines and geodesics
- Euler's Methodus Inveniendi and the two Additamenta
Finally, the authors give the readers a sense of how vast the calculus
of variations has become in centuries hence, providing some idea of what
lies outside the scope of the book as well as the current state of
affairs in the field.
This book will be of interest to anyone studying the calculus of
variations who wants a deeper intuition for the techniques and ideas
that are used, as well as historians of science and mathematics
interested in the development and evolution of modern calculus and
analysis.
