Change and Variations: A History of Differential Equations to 1900 (Springer Undergraduate Mathematics Series)

This book presents a history of differential equations, both ordinary and partial, as well as the calculus of variations, from the origins of the subjects to around 1900. Topics treated include the wave equation in the hands of d’Alembert and Euler; Fourier’s solutions to the heat equation and the contribution of Kovalevskaya; the work of Euler, Gauss, Kummer, Riemann, and Poincaré on the hypergeometric equation; Green’s functions, the Dirichlet principle, and Schwarz’s solution of the Dirichlet problem; minimal surfaces; the telegraphists’ equation and Thomson’s successful design of the trans-Atlantic cable; Riemann’s paper on shock waves; the geometrical interpretation of mechanics; and aspects of the study of the calculus of variations from the problems of the catenary and the brachistochrone to attempts at a rigorous theory by Weierstrass, Kneser, and Hilbert. Three final chapters look at how the theory of partial differential equations stood around 1900, as they were treated by Picard and Hadamard. There are also extensive, new translations of original papers by Cauchy, Riemann, Schwarz, Darboux, and Picard. The first book to cover the history of differential equations and the calculus of variations in such breadth and detail, it will appeal to anyone with an interest in the field. Beyond secondary school mathematics and physics, a course in mathematical analysis is the only prerequisite to fully appreciate its contents. Based on a course for third-year university students, the book contains numerous historical and mathematical exercises, offers extensive advice to the student on how to write essays, and can easily be used in whole or in part as a course in the history of mathematics. Several appendices help make the book self-contained and suitable for self-study. Preface The Shape of the Book Advice to Students Historiographical Remarks What This Book is Not Sources and Their Uses Advice to Instructors Acknowledgements Topics Discussed in This Book Contents List of Figures 1 The First Ordinary Differential Equations 1.1 Introduction 1.2 Origins: Inverse Tangent Problems 1.2.1 Debeaune's Problem 1.2.2 Other Inverse Tangent Problems 1.3 From Inverse Tangent Problems to Differential Equations 1.4 Differential Equations 1.5 Linear Ordinary Differential Equations 1.5.1 A Note on the Adjoint Equation 1.6 Exercises 2 Variational Problems and the Calculus 2.1 Introduction 2.2 Bernoulli's Problems 2.3 The Bernoullis' Brachistochrones 2.4 Geodesics on Surfaces 2.5 Exercises 3 The Vibrating String and the Partial Differential Calculus 3.1 Introduction 3.2 Early Investigations into the Partial Differential Calculus 3.3 D'Alembert: The Vibrating String and the Wave Equation 3.3.1 D'Alembert's Breakthrough 3.3.2 Mersenne's Law and Modes 3.4 Euler Rewrites the Wave Equation 3.5 Formal Complex Methods 3.6 Exercises 4 Rational Mechanics 4.1 Introduction 4.2 Fluid Mechanics 4.2.1 Recent Discoveries About the Euler Equations 4.3 Euler and the Propagation of Sound 4.4 Euler's Vision of Mechanics 4.5 Darboux's Account 4.6 Exercises 5 The Early Theory of Partial Differential Equations 5.1 Introduction 5.2 Euler's General Theory of Partial Differential Equations 5.2.1 Second-Order Partial Differential Equations 5.3 The Introduction of Characteristics by d'Alembert 5.4 Laplace 5.4.1 Lagrange's Method 5.5 Exercises 6 Lagrange's General Theory of Partial Differential Equations 6.1 Introduction 6.2 Clairaut's Paradox 6.3 Lagrange 6.3.1 Lagrange Lagrange1774 6.3.2 Lagrange Lagrange1774 6.3.3 Lagrange Lagrange1779 6.4 Exercises 7 The Calculus of Variations 7.1 Introduction 7.2 The Euler–Lagrange Equations Discovered 7.3 Maupertuis and the Principle of Least Action 7.4 Euler's Later Approach 7.5 Brachistochrone and the Calculus of Variations 7.6 Generalised Coordinates 7.7 Exercises 8 Monge and Solutions to Partial Differential Equations 8.1 Introduction 8.2 Monge and First-Order Partial Differential Equation 8.2.1 A Comparison with the Modern Account 8.2.2 The General First-Order Case 8.3 Monge on General First-Order Equation 8.4 Monge on Second-Order Partial Differential Equation 8.5 Lagrange at the École Polytechnique, 1806 8.5.1 Lacroix's Traité (1798) 8.6 Exercises 9 Revision 9.1 Revision and Assessment 1 9.1.1 Comments 10 The Heat Equation 10.1 Introduction 10.2 Fourier and His Series 10.2.1 Dirichlet on the Convergence of Fourier Series 10.2.2 Fourier Integrals 10.3 The Analysis of Fourier Integrals 10.4 Stokes and Laplace Transform 10.5 Exercises 11 Gauss and the Hypergeometric Equation 11.1 Introduction 11.2 Elliptic Integrals 11.3 Gauss 11.3.1 The Hypergeometric Equation 11.4 Kummer and His 24 Solutions 11.5 The Method of Undetermined Coefficients 11.6 Exercises 12 Existence Theorems 12.1 Introduction 12.2 Cauchy and Ordinary Differential Equations 12.2.1 Later Developments 12.3 Exercises 13 Riemann and Complex Function Theory 13.1 Introduction 13.2 Complex Function Theory 13.3 The Riemann Mapping Theorem 13.4 A Look Ahead 13.5 Exercises 14 Riemann and the Hypergeometric Equation 14.1 Introduction 14.1.1 Ordinary Differential Equations and Many-Valued Functions 14.1.2 The Riemann Sphere 14.2 Riemann's P-Functions 14.3 Riemann's Arguments 14.4 Exercises 15 Schwarz and the Complex Hypergeometric Equation 15.1 Introduction 15.2 Quotients of Solutions 15.3 Exercises 16 Complex Ordinary Differential Equations: Poincaré 16.1 Introduction 16.2 Poincaré and Linear Ordinary Differential Equations 16.3 Poincaré's Breakthrough and Non-Euclidean Geometry 16.4 Non-Euclidean Geometry 16.4.1 Summary 16.5 Exercises 17 More General Partial Differential Equations 17.1 Introduction 17.2 Cauchy's Method in 1819 17.3 Cauchy and the General Partial Differential Equation 17.4 Kovalevskaya's Theorem and Her Counter-Example 17.5 Exercises 18 Green's Functions and Dirichlet's Principle 18.1 Introduction 18.2 Green's Theorems and Green's Functions 18.3 Dirichlet Principle and Problem 18.4 Riemann on Green's Theorem 18.5 Riemann on the Dirichlet Principle 18.6 Exercises 19 Attempts on Laplace's Equation 19.1 Introduction 19.2 Weierstrass, Prym, and Schwarz 19.2.1 Schwarz's Alternating Method 19.3 Harnack 19.4 Exercises 20 Applied Wave Equations 20.1 Introduction 20.2 The Trans-Atlantic Cable 20.3 Poincaré's Solution 20.3.1 Conclusion 20.4 Exercises 21 Revision 21.1 Revision and Assessment 2 22 Riemann's Shockwave Paper 22.1 Introduction 22.2 Riemann's Paper 22.3 Darboux on Riemann's Approach to the Shockwave Equation 22.4 Telegraphy 22.5 Exercises 23 The Example of Minimal Surfaces 23.1 Introduction 23.2 Euler and Lagrange 23.3 Meusnier, Monge, and Legendre 23.4 Riemann and Weierstrass 23.5 Simple Solutions of the Plateau Problem 23.6 Exercises 24 Partial Differential Equations and Mechanics 24.1 Introduction 24.2 Hamiltonian Dynamics 24.3 Hamilton's and Jacobi's Theories of Dynamics 24.3.1 Jacobi 24.4 First-Order Partial Differential Equation Theory 24.5 Exercises 25 Geometrical Interpretations of Mechanics 25.1 Introduction 25.2 Gaussian Curvature 25.2.1 Liouville's Contributions 25.3 Geometrising Mechanics 25.4 The Connection to Hamilton–Jacobi Theory 25.5 Exercises 26 The Calculus of Variations in the nineteenth Century 26.1 Introduction 26.2 After Lagrange 26.3 Weierstrass's Theory 26.3.1 Two Examples 26.4 Hilbert's Problem 23 and the Theory of the Calculus of Variations 26.5 Exercises 27 Poincaré and Mathematical Physics 27.1 Introduction 27.2 The Classical Classification of Linear Partial Differential Equations 27.3 Poincaré and the Dirichlet Problem 27.4 Exercises 28 Elliptic Equations and Regular Variational Problems 28.1 Introduction 28.2 Picard on Second-Order Linear Elliptic Equations 28.3 Hilbert's Problems 19 and 20 28.4 Exercises 29 Initial Value Conditions for Hyperbolic Partial Differential Equations 29.1 Introduction 29.2 Picard on Second-Order Linear Hyperbolic Equations 29.3 Hadamard and Mathematical Physics 29.4 The Cauchy Problem 29.4.1 Commentary and Concluding Remarks 29.5 Exercises 30 Revision 30.1 Revision and Assessment 3 31 Translations 31.1 Cauchy: Note on the Integration of First-Order Partial Differential ... 31.2 Riemann's Lectures on Partial Differential Equations and Physics 31.2.1 Riemann, Introduction to Partial Differential Equations 31.3 Extracts from Schwarz, ``Ueber eine Abbildungsaufgaben'', 1869 31.3.1 The Schwarz–Christoffel Transformation 31.4 An Extract from Schwarz, On the Alternating Method 31.5 Schwarz on the Hypergeometric Equation (1873)—A Summary 31.6 Darboux on the Solution of Riemann's Equation (1887) 31.7 Picard and Elliptic Partial Differential Equations (1890) 31.8 Picard and Hyperbolic Partial Differential Equations (1890) Appendix A Newton's Principia Mathematica A.1 Newton's Laws of Motion in His Principia A.1.1 The Content of the Principia A.2 The Motion of the Moon Appendix B Characteristics B.1 First-Order Linear Partial Differential Equations B.2 Burgers' Equation, a Non-linear Equation Appendix C The First-Order Non-linear Partial Differential Equation Appendix D Green's Theorem and Heat Conduction D.1 Explicit Representations D.1.1 Adjoint Equations D.1.2 Boundary Value Problems Appendix E Complex Analysis E.1 Harmonic Functions E.2 Branch Points and Many-Valued ``Functions'' E.3 Analytic Continuation E.4 Liouville's Theorem Appendix F Möbius Transformations F.1 Möbius Transformations F.2 Inversion in a Circle F.2.1 Maps of the Unit Disc to Itself F.2.1.1 Coaxial Circles Appendix G Lipschitz and Picard G.1 Picard's Method Appendix H The Assessment H.1 Introduction H.2 Assessment 1 H.3 Assessment 2 H.3.1 Cauchy H.3.2 Thomson H.3.3 The Hypergeometric Equation H.3.4 Schwarz H.4 Assessment 3 H.4.1 Advice H.4.2 How the Essays Will Be Marked Appendix References Index