We begin our tour of useful mathematics with what is called the calculus of variations. Many physics problems can be formulated in the language of this calculus, and once they are there are useful tools to hand. In the text and associated exercises we will meet some of the equations whose solution will occupy us for much of our journey.

What is it good for?

The classical problems that motivated the creators of the calculus of variations include:

1. (i) Dido's problem: In Virgil's Aeneid, Queen Dido of Carthage must find the largest area that can be enclosed by a curve (a strip of bull's hide) of fixed length.

2. (ii) Plateau's problem: Find the surface of minimum area for a given set of bounding curves. A soap film on a wire frame will adopt this minimal-area configuration.

3. (iii) Johann Bernoulli's brachistochrone: A bead slides down a curve with fixed ends. Assuming that the total energy ½ mv2 + V(x) is constant, find the curve that gives the most rapid descent.

4. (iv) Catenary: Find the form of a hanging heavy chain of fixed length by minimizing its potential energy.

These problems all involve finding maxima or minima, and hence equating some sort of derivative to zero. In the next section we define this derivative, and show how to compute it.

Functionals

In variational problems we are provided with an expression J[y] that “eats” whole functions y(x) and returns a single number. Such objects are called functionals to distinguish them from ordinary functions.