## Chaotic Pendulum

### coupled oscillations - non-linear equations - chaos

** What it shows: **

A double pendulum executes simple harmonic motion (two normal modes) when
displacements from equilibrium are small. However, when large displacements
are imposed, the non-linear system becomes dramatically chaotic in its motion
and demonstrates that deterministic systems are not necessarily predictable.

** How it works: **

The double pendulum consists of two physical pendulums, each free to rotate a
full 360° around its pivot. The pivots are 5/8" diam. ball-bearings. The two
arms of the upper pendulum are fabricated from 1/4" thick aluminum and the
lower pendulum from 1/2" thick aluminum. The pendulum lengths are approximately
10.75" and their masses are equal. The general design was adapted from an MIT
version provided to us by Jack Wisdom (see reference below for more details).
The Lagrangian of the system yields two coupled nonlinear second-order differential
equations in the variables θ_{1} and θ_{2}, too long to reproduce here.
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1
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For small angles these equations become linear, with solutions that are linear
combinations of the two normal modes of vibration. The two normal modes can
easily be set into motion and demonstrated. The period of the symmetric mode
in which the arms oscillate in the same direction is 1.2 sec. The antisymmetric
mode (in which the upper and lower pendulums move in opposite directions) period
is, of course, shorter and equal to 0.5 sec. Increasing the initial displacement
to large angles results in chaotic motion which is quite fascinating and entertaining
to watch. The pendulums rotate clockwise and counterclockwise in the most unexpected
and surprising fashion, changing directions or suddenly increasing speed just when
you think you know what they're going to do next.

The double pendulum can be set up with fairly reproducible initial conditions.
For example, it can be fixed with a support arm so that it starts off with an
initial angular displacement of 90°. But regardless how carefully one tries to
secure identical releases from this initial configuration, the resultant motion
is never the same. Any arbitrarily small change in the initial conditions results
in a large change in the subsequent motion as these small deviations are amplified
exponentially in time. This is one of the salient features of chaotic phenomena
and it means that even "noise" in the initial conditions will be greatly amplified
with time. Released as described above, our pendulum executes chaotic motion for
about 1 minute and then settles down to nearly predictable motion for the next 8 or
9 minutes.

** Setting it up: **

The double pendulum support arm must be rigidly clamped to the edge of the lecture
bench. Unnecessary vibrations in the mount will usurp its energy and it will not
swing as long. It can be illuminated with a UV light for added effect (the
fluorescent tape on it glows brilliantly in the darkened lecture hall).

** Comments: **

We cannot pass up this opportunity to contrast our world view with the
*Weltanschauung* of two mathematicians who consider the complexity of their
equations to be the ultimate beauty. In the introductory paragraph of their paper
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they write, "As far as we know the experiment has never been performed but that is
not important...in fact, it may not even be wise to do the actual experiment in
order to understand the double pendulum...given that numerical computation is by
now much faster than classical experimentation - why should anybody put an effort
in the real thing?" Our mission is to inspire people's appreciation of the workings
of the physical world with the real thing. And this thing is nifty. Rating***

1
see, for example, T. Shinbrot, C. Grebogi, J. Wisdom, and J.A. Yorke, Am J Phys
**60**, 491 (1992). "Chaos in a double pendulum" (a good introduction to
and explanation of chaotic behavior)

J. Blackburn and G. Baker have investigated in detail and compared commercially
available chaotic pendulums. They report their findings in Am J Phys **66**(9),
pp 821-830 (1998). This apparatus note will be useful to people who don't have
the resources or desire to make their own. You will also find many good references
for the theory in this note.

2
Peter.H. Richter and Hans-Joachim Scholz, "Chaos in Classical Mechanics: The Double
Pendulum," pp 86-97, Fachbereich Physik der Universität Bremen, D-2800 Bremen 33,
Fed. Rep. of Germany.

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