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Phase Space of a Simple Pendulum

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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.
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