1 - Introductory example: a gravitational catastrophe machine

Summary

‘I am afraid that I rather give myself away when I explain,’ said he. ‘Results without causes are much more impressive.’

(The Stockbroker's Clerk)

It is well established that one should never begin a talk – or presumably a book – with an apology. We apologize, therefore, for apologizing that despite the title of this chapter our book is not primarily about catastrophe theory. The reason for our beginning with a gravitational catastrophe machine is that it exemplifies, in a vivid way, many of the ideas we shall study in detail later, such as functions on a curve, envelopes, surfaces, projections, evolutes and bifurcation sets. These ideas are merely touched on in the present chapter: do not expect to understand all the details yet.

The gravitational catastrophe machine was invented by T. Poston and is discussed in the well-known book on the subject (Poston and Stewart, 1978). Other introductions to catastrophe theory can be found in Zeeman (1977), Poston and Stewart (1976), Saunders (1980).

Consider a parabola, cut off by a line (perpendicular to the axis say), as in fig. 1.1. Imagine the region enclosed to be a lamina (thin sheet) that is constrained to move in a vertical plane, resting on a horizontal line; we seek the position of stable equilibrium. We do not assume the lamina to be of uniform density; in fact let its centre of gravity be at the point (a, b) referred to axes x and y as shown, relative to which the equation of the parabola is y = x².