The calculus of variations / Bruce van Brunt.

Format:

Book

Author/Creator:

Van Brunt, B. (Bruce)

Series:

Universitext

Language:

English

Subjects (All):

Calculus of variations.

Physical Description:

xiii, 290 pages : illustrations ; 25 cm.

Place of Publication:

New York : Springer, [2004]

Summary:

The calculus of variations has a long history of interaction with other branches of mathematics, such as geometry and differential equations, and with physics, particularly mechanics. More recently, the calculus of variations has found applications in other fields, such as economics and electrical engineering. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations. This book is an introductory account of the calculus of variations suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering. The mathematical background assumed of the reader is a course in multivariable calculus, and some familiarity with the elements of real analysis and ordinary differential equations.

The book focuses on variational problems that involve one independent variable. The fixed endpoint problem and problems with constraints are discussed in detail. In addition, more advanced topics, such as the inverse problem, eigenvalue problems, separability conditions for the Hamilton-Jacobi equation, and Noether's theorem, are discussed. The text contains numerous examples to illustrate key concepts along with problems to help the student consolidate the material. The book can be used as a textbook for a one-semester course on the calculus of variations, or as a book to supplement a course on applied mathematics or classical mechanics.

Contents:

1 Introduction 1

1.1 Introduction 1

1.2 The Catenary and Brachystochrone Problems 3

1.2.1 The Catenary 3

1.2.2 Brachystochrones 7

1.3 Hamilton's Principle 10

1.4 Some Variational Problems from Geometry 14

1.4.1 Dido's Problem 14

1.4.2 Geodesics 16

1.4.3 Minimal Surfaces 20

1.5 Optimal Harvest Strategy 21

2 The First Variation 23

2.1 The Finite-Dimensional Case 23

2.1.1 Functions of One Variable 23

2.1.2 Functions of Several Variables 26

2.2 The Euler-Lagrange Equation 28

2.3 Some Special Cases 36

2.3.1 Case I: No Explicity y Dependence 36

2.3.2 Case II: No Explicit x Dependence 38

2.4 A Degenerate Case 42

2.5 Invariance of the Euler-Lagrange Equation 44

2.6 Existence of Solutions to the Boundary-Value Problem* 49

3 Some Generalizations 55

3.1 Functionals Containing Higher-Order Derivatives 55

3.2 Several Dependent Variables 60

3.3 Two Independent Variables* 65

3.4 The Inverse Problem* 70

4 Isoperimetric Problems 73

4.1 The Finite-Dimensional Case and Lagrange Multipliers 73

4.1.1 Single Constraint 73

4.1.2 Multiple Constraints 77

4.1.3 Abnormal Problems 79

4.2 The Isoperimetric Problem 83

4.3 Some Generalizations on the Isoperimetric Problem 94

4.3.1 Problems Containing Higher-Order Derivatives 95

4.3.2 Multiple Isoperimetric Constraints 96

4.3.3 Several Dependent Variables 99

5 Applications to Eigenvalue Problems* 103

5.1 The Sturm-Liouville Problem 103

5.2 The First Eigenvalue 109

5.3 Higher Eigenvalues 115

6 Holonomic and Nonholonomic Constraints 119

6.1 Holonomic Constraints 119

6.2 Nonholonomic Constraints 125

6.3 Nonholonomic Constraints in Mechanics* 131

7 Problems with Variable Endpoints 135

7.1 Natural Boundary Conditions 135

7.2 The General Case 144

7.3 Transversality Conditions 150

8 The Hamiltonian Formulation 159

8.1 The Legendre Transformation 160

8.2 Hamilton's Equations 164

8.3 Symplectic Maps 171

8.4 The Hamilton-Jacobi Equation 175

8.4.1 The General Problem 175

8.4.2 Conservative Systems 181

8.5 Separation of Variables 184

8.5.1 The Method of Additive Separation 185

8.5.2 Conditions for Separable Solutions* 190

9 Noether's Theorem 201

9.1 Conservation Laws 201

9.2 Variational Symmetries 202

9.3 Noether's Theorem 207

9.4 Finding Variational Symmetries 213

10 The Second Variation 221

10.1 The Finite-Dimensional Case 221

10.2 The Second Variation 224

10.3 The Legendre Condition 227

10.4 The Jacobi Necessary Condition 232

10.4.1 A Reformulation of the Second Variation 232

10.4.2 The Jacobi Accessory Equation 234

10.4.3 The Jacobi Necessary Condition 237

10.5 A Sufficient Condition 241

10.6 More on Conjugate Points 244

10.6.1 Finding Conjugate Points 245

10.6.2 A Geometrical Interpretation 249

10.6.3 Saddle Points* 254

10.7 Convex Integrands 257

A Analysis and Differential Equations 261

A.1 Taylor's Theorem 261

A.2 The Implicit Function Theorem 265

A.3 Theory of Ordinary Differential Equations 268

B Function Spaces 273

B.1 Normed Spaces 273

B.2 Banach and Hilbert Spaces 278.

Notes:

Includes bibliographical references (pages [283]-285) and index.

ISBN:

0387402470

OCLC:

52182605