First and Second Variations, and Dirichlet’s Principle

Introduction: What is Calculus of Variations?

The calculus of variations refers to the science (or the art) of solving optimization problems containing integrals. In other words, out of a collection of functions, choose the one that makes the integral of those functions as small as possible. Such a function will be called a minimizer. Sometimes the integrand will depend on the function itself alone; sometimes dependence on derivatives of the function will be allowed. Here is an abstract example of what one of these problems looks like:

This is an excerpt from Filip Rindler’s “Calculus of Variations” text, Springer Publications. Arguably my favorite text on the subject matter.

If you aren’t familiar with what the “W” means, don’t worry about it right now (this is related to the theory of Sobolev Spaces).

When you’re given one of these problems, there are really two distinct, but related, questions being asked in one bundle:

1. Does a minimizer of the problem exist?
2. If a minimizer exists, is it unique?

To answer these two questions about variational principles, a wide range of areas of mathematics may be called upon. Functional analysis will take up a large part of this, but partial differential equations also like to make appearances. The applications are found in many places. Many of the definitions and notions that were introduced in the calculus of variations were…