A History of the Calculus of Variations from the 17th through the 19th Century

16.12.1980

Englisch

9780387905211

Springer US

Studies in the History of Mathematics and Physical Sciences

Nein

H. H. Goldstine

410

Gebundene Ausgabe

5

Variations, History, Calculus of Variations, Isaac Newton, History of Mathematics, Calculus, Variationsrechnung, Fermat's principle, isoperimetric inequality

PBX - Geschichte der Mathematik PBK - Mathematische Analysis, allgemein PDX - Geschichte der Naturwissenschaften

1. Fermat, Newton, Leibniz, and the Bernoullis.- 1.1. Fermat’s Principle of Least Time.- 1.2. Newton’s Problem of Motion in a Resisting Medium.- 1.3. The Brachystochrone Problem.- 1.4. The Problem Itself.- 1.5. Newton’s Solution of the Brachystochrone Problem.- 1.6. Leibniz’s Solution of the Brachystochrone Problem.- 1.7. John Bernoulli’s First Published Solution and Some Related Work.- 1.8. James Bernoulli’s Solution.- 1.9. James Bernoulli’s Challenge to His Brother.- 1.10. James Bernoulli’s Method.- 1.11. John Bernoulli’s 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. Lagrange’s First Letter to Euler.- 3.2. Lagrange’s First Paper.- 3.3. Lagrange’s Second Paper.- 3.4. Legendre’s 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. Jacobi’s Paper of 1836.- 4.3. Excursus on Planetary Motion.- 4.4. V.-A. Lebesgue’s Proof.- 4.5. Hamilton-Jacobi Theory.- 4.6. Hesse’s Commentary.- 5. Weierstrass.- 5.1. Weierstrass’s 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. Scheeffer’s Results.- 5.12. Schwarz’s Proof of the Jacobi Condition.- 5.13. Osgood’s Summary.- 6. Clebsch, Mayer, and Others.- 6.1. Introduction.- 6.2. Clebsch’s Treatment of the Second Variation.- 6.3. Clebsch, Continuation.- 6.4. Mayer’s Contributions.- 6.5. Lagrange’s 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. Hilbert’s Invariant Integral.- 7.2. Existence of a Field.- 7.3. Hibert, Continuation.- 7.4. Mayer Families of Extremals.- 7.5. Kneser’s Methods.- 7.6. Kneser on Focal Points and Transversality.- 7.7. Bliss’s Work on Problems in Three Space.- 7.8. Boundary-Value Methods.- 7.9. Hilbert’s Existence Theorem.- 7.10. Bolza and the Problem of Bolza.- 7.11. Carathéodory’s Method.- 7.12. Hahn on Abnormality.

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