The Early Period of the Calculus of Variations

1. Front Matter

   Pages i-xii

   PDF

2. Some Introductory Material

   • Paolo Freguglia, Mariano Giaquinta

   Pages 1-37

3. The Brachistochrone Problem: Johann and Jakob Bernoulli

   • Paolo Freguglia, Mariano Giaquinta

   Pages 39-57

4. Isoperimetrical Problems: Jakob and Johann Bernoulli

   • Paolo Freguglia, Mariano Giaquinta

   Pages 59-99

5. Shortest Lines and Geodesics

   • Paolo Freguglia, Mariano Giaquinta

   Pages 101-121

6. Euler’s Memoirs of 1738 and 1741

   • Paolo Freguglia, Mariano Giaquinta

   Pages 123-153

7. Euler’s Methodus Inveniendi

   • Paolo Freguglia, Mariano Giaquinta

   Pages 155-191

8. Lagrange’s \(\delta \)–Calculus

   • Paolo Freguglia, Mariano Giaquinta

   Pages 193-278

9. Back Matter

   Pages 279-293

   PDF

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.

 

• Calculus of Variations
• Infinitesimals
• Johann Bernoulli
• Jakob Bernoulli
• Brachystochrone Problem
• Leonhard Euler
• Isoperimetric Problems
• Lagrange's delta-Calculus

“The book presents an accurate description of the ideas and methods developed to face the first variational problems, essentially from the Brachistochrone in the seventeenth century to the very first years of the nineteenth century, covering overall a little more than a century. … The book is the result of the fruitful cooperation between a specialist in the history of sciences and a well-known researcher in the Calculus of Variations … .” (Arrigo Cellina, Mathematical Reviews, May, 2017)



“The authors succeeded in adding a very valuable book to the literature available on the history of the calculus of variations. The book is full of the mathematics of the masters and the authors have taken pain to clarify even the details in nowadays mathematical notations.” (Thomas Sonar, zbMATH 1350.01001, 2017)

• Dept. of Pure and Applied Mathematics, Universit`a di L’Aquila, Porto Santo Stefano, Italy

  Paolo Freguglia

• Scuola Normale Superiore, Pisa, Italy

  Mariano Giaquinta

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