Introduction

The calculus of variations is a branch of mathematics which considers extremal problems; it yields techniques for determining when a particular definite integral will be a maximum or a minimum (or, more generally, the conditions for the integral to be “stationary”). The calculus of variations answers questions such as the following.

1. • What is the path that gives the shortest distance between two points in a plane? (A straight line.)

2. • What is the path that gives the shortest distance between two points on a sphere? (A geodesic or “great circle.”)

3. • What is the shape of the curve of given length that encloses the greatest area? (A circle.)

4. • What is the shape of the region of space that encloses the greatest volume for a given surface area? (A sphere.)

The technique of the calculus of variations is to formulate the problem in terms of a definite integral, then to determine the conditions under which the integral will be maximized (or minimized). For example, consider two points (P1 and P2)inthe x–y plane. These can be connected by an infinite number of paths, each described by a function of the form y = y(x). Suppose we wanted to determine the equation y = y(x) for the curve giving the shortest path between P1 and P2.