The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. Although it is now at the core of many modern mathematical fields, it does not have a well-defined place in most undergraduate mathematics curricula. This volume should nevertheless give the undergraduate reader a sense of its great character and importance.
Interesting functionals, such as area or energy, often give rise to problems whose most natural solution occurs by differentiating a one-parameter family of variations of some function. The critical points of the functional are related to the solutions of the associated Euler-Lagrange equation. These differential equations are at the heart of the calculus of variations and its applications to other subjects. Some of the topics addressed in this book are Morse theory, wave mechanics, minimal surfaces, soap bubbles, and modeling traffic flow. All are readily accessible to advanced undergraduates.
This book is derived from a workshop sponsored by Rice University. It is suitable for advanced undergraduates, graduate students and research mathematicians interested in the calculus of variations and its applications to other subjects.
Author(s): Robin Forman, Frank Jones, Barbara Lee Keyfitz, Frank Morgan, Michael Wolf, Steven J. Cox, Robert Hardt
Series: Student Mathematical Library, V. 26
Publisher: AMS
Year: 2004
Language: English
Pages: 170
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
List of Contributors......Page 12
Calculus of Variations: What Does "Variations" Mean?......Page 14
II. The domain is n-dimensional......Page 15
IV. The Euler-Lagrange scenario......Page 16
A. Minimal surfaces. G......Page 19
B. Geodesics.......Page 21
C. Isoperimetric problem. T......Page 22
VI. Important disclaimer......Page 24
How Many Equilibria Are There? An Introduction to Morse Theory......Page 26
I. Dynamical systems......Page 27
II. Topology......Page 35
III. Morse Theory......Page 38
V. The idea of the proof......Page 41
VI. What does it really mean to build a shape from cells?......Page 43
VII. What now?......Page 46
1. Introduction......Page 50
2. Acquiring the data......Page 51
3. A mathematical model......Page 56
4. Solving the wave equation......Page 60
5. The damped wave equation......Page 62
6. Discerning the presence of additionaldamping......Page 66
7. Concluding remarks......Page 68
Bibliography......Page 69
1. The news.......Page 72
3. Proof of the Planar Double Bubble Theorem.......Page 73
4. The standard double bubble. For......Page 77
6. Symmetry Theorem. A......Page 78
7. Monotonicity [12, THM. 3.2].......Page 79
9. Hutchings Structure Theorem [12, THM. 5.1].......Page 80
10. Hutchings component bound. C......Page 81
12. Theorem ([7], [9]). F......Page 83
13. Theorem [11].......Page 85
14. Higher dimensions.......Page 87
Bibliography......Page 89
Minimal Surfaces, Flat Cone Spheres and Moduli Spaces of Staircases......Page 92
1. Minimal surfaces......Page 93
2. Some history......Page 97
3.1. Direct method. W......Page 99
3.2. Intrinsic vs. extrinsic geometry. T......Page 100
3.3. Riemann surfaces. O......Page 101
3.4. Some real differential geometry. W......Page 102
3.5. Digression on complex analysis. It......Page 104
3.6. From real differential geometry to complex analysis.......Page 107
3.7. The Weierstrass representation.......Page 112
3.8. Examples. A......Page 115
3.9. Some restrictions. A......Page 116
4.1. The problem and some history. W......Page 119
4.2. The proof.......Page 123
References......Page 137
1. Introduction: A continuum model for traffic flow......Page 140
2. Some conservation law theory......Page 145
3. An application of the model: The timing of traffic lights......Page 155
4. Extensions and other models......Page 160
Bibliography......Page 164
Back Cover......Page 170
