35.1 The Undamped Pendulum

In Chapter 13 we derived the exact equation for the motion of an ideal pendulum,

(where ? ² = g/L) but we then approximated this by = ? ? ² ? in order to apply the methods we had just learned for linear equations.

Here we will use phase plane methods to understand the nonlinear equation (35.1). For simplicity we will choose ? = 1 and consider the equation

In order to look at this as a set of coupled first order equations we set x = ? and y = and then

Note that the direction field (shown in Figure 35.1) repeats itself every 2 ? in the horizontal direction. This should not be a surprise, since the x coordinate represents the angle of the pendulum to the vertical ( ? in our original equation), and the value ? = x + 2 ? corresponds to the same position of the pendulum as ? = x. So we should consider ( x, y) and ( x + 2 ? , y) as representing the same physical state of the system. The natural way to present our phase diagrams, then, is to restrict to a range of x corresponding to one particular choice for the angle ?, ?