1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 483

Physically the critical points (2mr, 0) correspond to the pendulum being at

rest (y2 = y' = 0) and hanging vertically downward from the support S,

(Y1 = y = 2mr). Likewise, the physical interpretation of the critical points

((2n + l)7r, 0) is that the pendulum is at rest (y2 = 0) balanced vertically

above the support S, (y1 = (2n + l)7r). From these physical considera-

tions we deduce that the critical points (2n7r, 0) are stable and the critical

points ((2n + l)7r, 0) are unstable. Displayed in Figure 10.23 is a phase-

plane graph of Y2 versus Y1 for system (6) for the initial conditions y 1 (0) = 0

and (i) Y2(0) = 1.5, (ii) Y2(0) = 2, (iii) Y2(0) = 2.5, (iv) y2(0) = -2, and

(v) Y2(0) = -2.5. From this figure we see that if the pendulum is at rest in a

vertically downward position at time t = 0 and is struck sharply from the left

side imparting an angular velocity of 1.5 (y'(O) = 1.5), then the pendulum

executes periodic motion (harmonic motion) about the stable critical point

(0, 0). If the initial angular velocity is 2, then the pendulum swings to the

right (y' (0) = 2 > 0) and balances itself vertically above the support. That

is, the solution approaches the unstable critical point (7r, 0). (Of course, this
is not apt to happen in any laboratory experiment!) If the initial angular
velocity is 3, then the pendulum rotates indefinitely in a counterclockwise di-
rection about the support. The pendulum has the smallest angular velocity
when it is vertically above the support and it has the largest angular veloc-
ity when it is vertically below the support. When y 2 (0) = -2 the pendulum
swings to the left and balances itself above the support. And when y 2 (0) = -3
the pendulum rotates indefinitely in a clockwise direction about the support.
Compare these results with those obtained previously when the linear ap-
proximation sin y ~ y was made----that is, compare Figures 10.22 and 10.23.

4

2

y

2

0

-2

-4

-10 -5 0 5 10

Y1

Figure 10.23 Phase-Plane Portrait for System (6)

Simple Pendulum with Damping but N o Forcing Function

If we assume there is a damping force due to air resistance or friction at
the point of suspension which is proportional to the angular velocity (c =f. 0)