Phase Space of a Simple Pendulum

pendulum length

5.1

Consider the motion of a pendulum of length

l

described by the differential equation

2

d

θ

d

2

t

+

g

l

sin(θ)=0

, where

g

is the gravitational acceleration and

θ

is the angle between the pendulum and the vertical direction. This Demonstration plots the phase space diagram (i.e.,

θ

along the horizontal axis and

dθ

dt

on the vertical axis). The separatrix, plotted in red, is given by

H=

2



θ

2

-

g

l

cos(θ)=

g

l

, where

H

is the Hamiltonian of the system. The separatrix separates phase space into two different regions. The inside region, where the pendulum oscillates back and forth, corresponds to

-

g

l

<H<

g

l

. The outside region corresponds to

H>

g

l

, where the pendulum continuously turns through vertical planar circles.