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 for which the 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.
Steven J. Cox: Rice University, Houston, TX Robin Forman: Rice University, Houston, TX Frank Jones: Rice University, Houston, TX Barbara Lee Keyfitz: The Fields Institute, Toronto, ON, Canada Frank Morgan: Williams College, Williamstown, MA Michael Wolf: Rice University, Houston, TX
Edited by: Robert Hardt: Rice University, Houston, TX
Articles Frank Jones — 1. Calculus of variations: What does “variations” mean? Robin Forman — 2. How many equilibria are there? An introduction to Morse theory Steven J. Cox — 3. Aye, there’s the rub. An inquiry into why a plucked string comes to rest Frank Morgan — 4. Proof of the double bubble conjecture Michael Wolf — 5. Minimal surfaces, flat cone spheres and moduli spaces of staircases Barbara Lee Keyfitz — 6. Hold that light! Modeling of traffic flow by differential equations