Calculus of Variations Reading Course
Contents |
Introduction
In previous years, I have offered a course at SFU on Calculus of Variations, using the textbook
- Calculus of Variations with applications to physics and engineering by Robert Weinstock google books link
The textbook is a classical approach to the Calculus of Variations, which is now somewhat old-fashioned. Another book which is similar in style is
- An Introduction to the Calculus of Variations By Charles Fox google books link
More modern approaches can be found in
- Introduction to the calculus of variations By Bernard Dacorogna google books link
- Direct methods in the calculus of variations By Enrico Giusti google books link
The classical approach to the calculus of variations involves somewhat mysterious calculations, which are better digested given a very strong background in the computational aspects of calculus and differential equations. In fact, this requires a stronger background than is provided by modern undergraduate degrees which emphasize other topics. As a result some of the questions in Weinstock are either tediously long computations, or require knowledge of obscure special functions. On the other hand, the modern approach is very abstract - to the point where a reader may have no idea what the goals are, and very analytical. The second text is better approached as an application of analysis, rather than a source for solving applied problems.
The applications of the Calculus of Variations go back the the beginnings of calculus. There are a handful of classical problems, e.g.
- the brachistochrone wikipedia link
- the isoperimetric problem (also known as dido's problem) wikipedia link
- minimal surfaces wikipedia link
which are fun and an important part of mathematics.
Books on Mechanics and Calculus of Variations
- The Variational Principles of Mechanics By Cornelius Lanczos http://books.google.com/books?id=ZWoYYr8wk2IC
- Classical mechanics By John Robert Taylor http://books.google.com/books?id=P1kCtNr-pJsC
Active approach to learning the material
Part of what make the topic difficult to learn is that the classical books follow a historical approach, rather than a conceptual approach. In fact, to properly learn the material, it helps to review calculus, in particular
- minimization of functions of several variables
- constrained minimization (Lagrange Multipliers)
The latter topic comes up in the isoperimetric problem, where the curve is constrained to have a finite perimeter.
Personally, I find the whole topic more interesting in the context of more general kinds of minimization. A good book which has an overview of several topics is
- Introduction to Optimization By Pablo Pedregal google books link
which starts with opitmization, then has a chapter on Calculus of Variations, and chapters on
- dynamic programming
- optimal control
These are very interesting topics as well, and it's great to find a book which collects these topics together.
Digression on Dynamic Programming versus Calculus of Variations
While I'm on this subject, it may be helpful to clarify the difference between dynamic programming (applied to finding optimal paths) and the optimal paths via calc of variations. For example, both approaches can be used to find geodesics. Calculus of variations assumes that the endpoints of the curves are fixed, and tries to find the shortest (or cheapest) path connecting them. Dynamics Programming assumes that one point is fixed, and tries to find the shortest path from all nearby points to the special point.
While it seems like a fairly small distinction, it makes a big difference.
- In the Calculus of Variations case we get ordinary differential equations for optimal curves.
- In the Dynamic Programming case we get Partial Differential Equations for the distance (or least cost) function.
The optimal paths can be read off of this function (although that's an extra step).