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Calculus of variations: a lunchbreak guide

The calculus of variations is a branch of mathematics that deals with finding functions that optimize systems. That is, it finds functions which describe shapes or surfaces that minimize time or distance, or maximize strength, among other things. Calculus of variations is also of importance in theoretical physics, where is it used, among many other things, to determine the equations of motion of particles subject to particular conditions.

Although the calculus of variations has its origins in the 18th century, it is still not widely taught, either to science of engineering students. Kreyszig's family bible-sized Advanced Engineering Mathematics gives it not so much as a passing mention in any of its twelve hundred pages. In addition, the language of the subject, and the concepts on which it depends, are likely to be somewhat unfamiliar, even to engineers with a strong mathematics background.

This article attempts to describe the basic principles of calculus of variations, and solve a simple variational problem, in a way that should be comprehensible to a person with high-school calculus and algebra. My intention is that it should be possible to read it in about an hour.

I wrote this article in the hope that it will be useful to people who might be considering studying calculus of variations, or have an inkling that they might need it in their work, but have no idea where to start. There are many simplifications, and some important results are assumed, rather than proved. However, I hope that it is sufficient to give at least a taste of the subject.

The article begins by explaining the fundamental terminology of the subject. The remainder of the article concentrates of solving a specific variational problem, the notorious brachistochrone problem. It's not the simplest of applications of calculus of variations — although it's pretty simple as these things go — but it seems to me to be the simplest non-trivial example. That is, it's the simplest example that produces a result which you probably couldn't intuit or derive from simple geometry. We will apply the Euler-Lagrange equation — one of the fundamental tools of the subject — to the physical system to produce a differential equation, and then solve this equation. Finally, we will see plots of the actual solution functions for specific geometries of the physical system.

Technical note: this article uses the MathJax library for formatting equations. This is a JavaScript library, and you might get script warnings from some browsers, particular Microsoft Internet Explorer. Some pages might take a while to display on some systems.

Functionals, and their minimization
Representing the physical problem as a functional
Minimizing the functional
Solving the differential equation
Applying the boundary conditions
Summary

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Copyright © 1994-2013 Kevin Boone. Updated Aug 17 2013