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: