1
Chaotic Double Pendulum
Jusuk Lee, Brandon Sim, Max Sun
Ms. Jensvold – Physics C AP Period 5
22 September 2009
Investigation
How does the motion of double pendulum differ based on different angles and height? In this
lab, we explore the motion of double pendulums. By differing initial conditions, we see that the
pendulum can exhibit either chaotic or nonchaotic behavior. By doing this, the different
characteristics of chaotic and nonchaotic motion are observed. The purpose of this lab is to
explore the motions of the double pendulum system and to make a hypothesis about the
mathematical nature of the system. Through techniques of classical mechanics, notably the
analysis of the system’s Lagrangian, we are able to find that the system is in fact chaotic;
however, it can be modeled precisely for a given set of initial conditions. In addition, with the
data as well as additional analysis, we give a set of boundaries for the chaotic motion of the
system based on its initial parameters.
Method
Materials Needed:
• Double pendulum kit from CIPT lending library (wheels, screw, bolts, washers, clamps, etc.)
• Protractor
• Photogates with software like VernierPro (Although many parts of the demonstration require only
the pendulum kit).
Experimental Work
Section 1
Question 1:
Two characteristics of the double pendulum’s chaotic motion are the initial angles Θ
1
and Θ
2
as
well as the period of the top pendulum.
Question 2:
The behavior of the double pendulum is chaotic for certain energies (which is dependent on the
initial angles of the upper and lower pendulums, respectively, Θ
1
and Θ
2
.) Above certain energy,
the behavior of the double pendulum is erratic and chaotic, depending largely on the initial
conditions Θ
1
and Θ
2.
Under this threshold energy, however, the behavior of a double pendulum
is the same as that of a normal pendulum.
Section 2
Question 3
:
Pendulums do follow Newton’s Laws of motion. In fact, the chaotic motion of the double
pendulum can be modeled by numerically integrating (RK4 or other methods) several ODE’s
(ordinary differential equations). However, with just classical Lagrangian mechanics, the