Project 3: Fun with Pendulums

h = .005, Initial Angle: π/2

h = .01, Initial Angle: π/2

For this set of initial conditions, the distance the pendulum
travels decreases as time goes on. The pendulum travels roughly half
the distance each time. Each set of initial conditions end up
reapeating the periodic behavior, so only 1 video is included.

Since the damping force always occurs against the direction
the pendulum travels, there is a consistant decline in the pendulum's
angle. At small angles, these differences can seem to be the same size,
but observing the angle size when the pendulum reaches it's maximum
height, we can quantitatively say that isn't true. The larger step size
(h = .01) accounts for the quicker movement of the pendulum.

Tame Behavior: Angle 1: 5pi/6 Angle 2: 5pi/6

Chaotic Behavior: Angle 1: 5pi/6 Angle 2: pi/4

Here, the motion of the double pendulum is partially periodic.
In general, through a single swing, the first pendulum's velocity
will slow down to zero half way through, then pick up again,
while the second pendulum will make a single fast swing. This
particular test does not completely demonstrate this phenomenon.

Here, the double pendulum's behavior can hardly be described
as periodic. The full revolutions of both rods generate severely chaotic
turns.