Author	Goldstine, Herman H. author
Title	A History of the Calculus of Variations from the 17th through the 19th Century [electronic resource] / by Herman H. Goldstine
Imprint	New York, NY : Springer New York, 1980
Connect to	http://dx.doi.org/10.1007/978-1-4613-8106-8
Descript	410 p. online resource

The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not the true moment of inception of the subject. It would not have been unreasonable if I had gone back to the set of isoperimetric problems considered by Greek mathematiยญ cians such as Zenodorus (c. 200 B. C. ) and preserved by Pappus (c. 300 A. D. ). I have not done this since these problems were solved by geometric means. Instead I have arbitrarily chosen to begin with Fermat's elegant principle of least time. He used this principle in 1662 to show how a light ray was refracted at the interface between two optical media of different densities. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem. There have been several other histories of the subject, but they are now hopelessly archaic. One by Robert Woodhouse appeared in 1810 and another by Isaac Todhunter in 1861

1. Fermat, Newton, Leibniz, and the Bernoullis -- 1.1. Fermatโs Principle of Least Time -- 1.2. Newtonโs Problem of Motion in a Resisting Medium -- 1.3. The Brachystochrone Problem -- 1.4. The Problem Itself -- 1.5. Newtonโs Solution of the Brachystochrone Problem -- 1.6. Leibnizโs Solution of the Brachystochrone Problem -- 1.7. John Bernoulliโs First Published Solution and Some Related Work -- 1.8. James Bernoulliโs Solution -- 1.9. James Bernoulliโs Challenge to His Brother -- 1.10. James Bernoulliโs Method -- 1.11. John Bernoulliโs 1718 Paper -- 2. Euler -- 2.1. Introduction -- 2.2. The Simplest Problems -- 2.3. More General Problems -- 2.4. Invariance Questions -- 2.5. Isoperimetric Problems -- 2.6. Isoperimetric Problems, Continuation -- 2.7. The Principle of Least Action -- 2.8. Maupertuis on Least Action -- 3. Lagrange and Legendre -- 3.1. Lagrangeโs First Letter to Euler -- 3.2. Lagrangeโs First Paper -- 3.3. Lagrangeโs Second Paper -- 3.4. Legendreโs Analysis of the Second Variation -- 3.5. Excursus -- 3.6. The Euler-Lagrange Multiplier Rule -- 4. Jacobi and His School -- 4.1. Excursus -- 4.2. Jacobiโs Paper of 1836 -- 4.3. Excursus on Planetary Motion -- 4.4. V.-A. Lebesgueโs Proof -- 4.5. Hamilton-Jacobi Theory -- 4.6. Hesseโs Commentary -- 5. Weierstrass -- 5.1. Weierstrassโs Lectures -- 5.2. The Formulation of the Parametric Problem -- 5.3. The Second Variation -- 5.4. Conjugate Points -- 5.5. Necessary Conditions and Sufficient Conditions -- 5.6. Geometrical Considerations of Conjugate Points -- 5.7. The Weierstrass Condition -- 5.8. Sufficiency Arguments -- 5.9. The Isoperimetric Problem -- 5.10. Sufficient Conditions -- 5.11. Scheefferโs Results -- 5.12. Schwarzโs Proof of the Jacobi Condition -- 5.13. Osgoodโs Summary -- 6. Clebsch, Mayer, and Others -- 6.1. Introduction -- 6.2. Clebschโs Treatment of the Second Variation -- 6.3. Clebsch, Continuation -- 6.4. Mayerโs Contributions -- 6.5. Lagrangeโs Multiplier Rule -- 6.6. Excursus on the Fundamental Lemma and on Isoperimetric Problems -- 6.7. The Problem of Mayer -- 7. Hilbert, Kneser, and Others -- 7.1. Hilbertโs Invariant Integral -- 7.2. Existence of a Field -- 7.3. Hibert, Continuation -- 7.4. Mayer Families of Extremals -- 7.5. Kneserโs Methods -- 7.6. Kneser on Focal Points and Transversality -- 7.7. Blissโs Work on Problems in Three Space -- 7.8. Boundary-Value Methods -- 7.9. Hilbertโs Existence Theorem -- 7.10. Bolza and the Problem of Bolza -- 7.11. Carathรฉodoryโs Method -- 7.12. Hahn on Abnormality

1. Mathematics
2. History
3. Mathematical analysis
4. Analysis (Mathematics)
5. Mathematics
6. History of Mathematical Sciences
7. Analysis
8. History of Science