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How Chaos Theory Works


Not So Certain After All: Dynamical Instability

By the end of the 19th century, scientists were becoming a little complacent. Newton's laws had proven to be extraordinarily robust, and everyone assumed they could solve any physical problem set before them. In addition to this sturdy mathematical foundation, astronomers were adding more information about Earth and its position in the solar system and beyond. An astronomical chart of 1900 would have displayed the eight principal planets, each in an elliptical orbit around the sun, as well as numerous satellites, the larger asteroids and a handful of comets. The same chart would have provided apparent magnitudes, orbital velocities, diameters and distances from the sun. In other words, it contained all of the information necessary to exploit Newton's equations and determine a future state of the planets.

In 1885, King Oscar II of Sweden and Norway offered a prize to anyone who could prove the stability of the solar system. It may have seemed like an unnecessary quest (after all, the solar system had obviously been stable for millions of years before the 1800s), but it had titillated scientists for years and, at the very least, it provided a means to demonstrate the power of classical mechanics. Several well-known mathematicians, including Leonhard Euhler, Joseph-Louis Lagrange and even Pierre-Simon Laplace himself, had tackled the problem before King Oscar's contest. A few managed to provide proofs of solar system stability, at least in short-term models. But no one had been able to prove, definitively, that the eight planets would stay in a bounded region of space for all time.

Enter Henri Poincaré, a French mathematician already known for innovative thinking before the contest attracted his attention. Instead of focusing on all planets and the sun simultaneously, Poincaré decided to limit his analysis to a smaller, simpler system -- two massive bodies orbiting one another around their common center of gravity while a much smaller body orbits them both. This is known as the n-body problem, which uses complex math to predict the motion of a group of celestial objects that interact gravitationally. That math usually involves differential equations -- equations that give the rate of change of a system as a function of its present state. But when Poincaré tried to describe the present state of the bodies in his simplified calculus, he discovered that small imprecisions -- rounding off a planet's mass, for example -- grew over time and became magnified at an alarming rate. Even when he shrunk the uncertainties in initial conditions to smaller and smaller values, the calculations still "blew up," producing enormous uncertainties in the final predictions. He concluded it was impossible to predict the future outcome of the solar system because the system itself was far too complex, filled with too many variables that could never be measured with absolute precision.

For his work, Poincaré won the contest. But his real accomplishment was to discover something known as dynamical instability, or chaos. It largely went unnoticed for another 70 years, until a meteorologist at the Massachusetts Institute of Technology (MIT) tried to use computers to improve weather forecasting.