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The 'three-body problem' became apparent when considering the motion of three bodies mutually attracted by only the force of gravity and described in Isaac Newton's 'Principia' in 1687 applied to the Earth, Sun and Moon.
200 years later Heinrich Bruns and Henri Poincaré showed that there were no general analytic equations which could predict the motion of three such bodies. The general pattern of motion is non-repeating and unpredictable.
In the 1960's computer models were made clearly demonstrating the non-repeating, chaotic behaviour of systems similar to the three-body problem.
The Double Pendulum
Another example approaching the classic three body problem is the double pendulum, where one rod is attached to another rod, and both are able to swing freely. The three points considered in a model of the double pendulum are:
one fixed pivot (black circle),
the end of the first rod (green circle)
the end of the second rod (blue circle).
The line of motion of the end of the second rod exhibits typically chaotic, non-repeating motion as shown.
Download HTML Double pendulum model
The Lorenz attractor
A famous example is the Lorenz attractor, which is not a three-body problem as such, but has three variables which, at successive moments in time, change by an amount that depends on each value of the three variables.
Graphically this is displayed as the Lorenz attractor
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