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Physics LibreTexts

9.17: Chaos and Determinism

Determinism is the idea that the state of a physical system can be predicted exactly. Newton's law of gravity gives the force between two masses separated in space. The equation is exact. If we could measure the exact masses and the separation, the force and the subsequent motion would be exactly determined. Of course, we know that observations have uncertainties. But, in principle, two objects in a gravitational embrace work like a perfectly predictable machine. The success of Newton's theory led to the metaphor of a "clockwork universe."

The gravity between two objects can be calculated exactly. What happens if you add just one more object? When three bodies orbit each other, we must use a separate equation to describe the force between each pair. There is no exact solution to this set of equations. If you had a powerful computer, you could calculate the orbits with a high degree of precision, but it would take an infinite amount of time to reach infinite precision. When dealing with the gravity of three of more objects, Newton's law is not deterministic. Newton would probably have been amused by the strong reaction of poets of the Romantic era to his theory; they considered his theory to be tyrannical because it robbed humans of free will. But that is not how gravity works. In any realistic situation, the motions of objects can not be predicted exactly from the starting conditions. The future does not flow rigidly from the past.

In the past few decades, scientists have gained a second insight about gravity. In any complex system (which might only be three objects!) a slight change in the starting conditions can give a large change in the final outcome. This is the idea of chaos. Nature is not deterministic; it can even be wildly unpredictable! 

Chaos is a new field of science, with many applications. Perhaps the most familiar example is the weather. You may wonder why weather forecasts are mostly useless more than four or five days ahead, when giant super computers and armadas of satellites are working on the problem. Weather involves motions on a wide range of scales, from dust devils to tornadoes to continent-sized hurricanes. A small effect on a small scale can become a large effect on a large scale. Meteorologist Edward Lorenz called this the "butterfly effect," where a butterfly stirring the air in New York can eventually affect the weather in London. This cascade of small changes to large effects even has a place in folklore:

Astropedia Image
Edward Lorenz. Click here for original source URL.

"For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, the battle was lost;
For want of a battle, the kingdom was lost!"

Chaos seems synonymous with disorder. But order and disorder can coexist in the simplest physical system. Even the motion of a classic timekeeping device like a pendulum is not strictly periodic. It displays some chaotic behavior. Or consider a dripping faucet. A careful experiment will show that the drops do not fall at exactly equal time intervals. It is too simplistic to think only in terms of periodic or random processes. Scientists have been able to use computers to simulate Jupiter's Great Red Spot as a vortex in a rotating flow. After many orbits, most of the flow has broken up in turbulence but a long-lived storm persists — an island or order in a sea of chaos. These ideas may even help us understand the emergence of life on Earth.