Chaos Theory
Will the earth, on its current course, crash into the sun in ten thousand years? … One hundred thousand years? … A million years? ...
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A few years ago I worked at the Goddard satellite tracking station in South Africa and drove an old Austin 150-A to work. The four-cylinder coupe had only three good cylinders but managed to putt-putt along without significant problems. It did, that is, until the maintenance chief at the station got wind of my "plight" and offered to fix the wounded cylinder. And this he did, most efficiently! After considerable time and effort on his part, the Austin began to hum like a racing car. And that was the cause of the ensuing trouble.
The car now ran like a blue streak, but it had so much power it began to self-destruct. First the tires started popping. The radiator went next, because it couldn't handle the new heat load. And finally the main bearing burned out, sending the car to its happy hunting ground.
The moral of this little story? When a system is in equilibrium, a jolt like a 33% increase in power can seriously disrupt its operation and drive it into instability. In this case disaster struck. Out of order came chaos. Even a tiny alteration can create havoc. That's the nature of chaos.
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Is there a possibility our solar system is unstable and that the earth may eventually veer out of its current benign path and swing wildly into the sun? Is the universe itself unstable?
We're dealing here with deterministic systems -- ordinary physics. In principle, for such systems, the future is completely determined by the past -- i.e., it is completely calculable. So if our mechanical, Newtonian model of the solar system is correct, we should be able to predict the behavior of the system accurately for any time in the future, given present conditions.
Unfortunately, there is no known formula to plug in a future time and derive the state of the planets at that time. So we can't deduce the behavior of Earth from such equations.
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There are several ways to approach the n-body problem. For instance, we can simplify the conditions and thus simplify the equations. We can also use simulation.
Simplify the Equations
One option is to ignore the interactive effects of all but the most relevant objects in the n-body system. For example, the equation expressing the motion of the earth around the sun can be "solved" by ignoring the effects of all the other planets. In this event, n becomes 2, and we have a soluble two-body problem.
Having been derived from a simplified model, however, the solution only approximates the true answer. The fact is, we have no way of knowing the accuracy of the answer, because we don't know precisely how the interactions with the other planets affect the answer. To learn their effects, we must include this data.
Use Simulation
If we insist on taking the whole system into account, the best we can do at present is to use simulation to generate future states. Following this procedure, we would include the equations of motion for all of the planets in the solar system and their gravitational interactions and spell out the gravitational influences among them.
By grinding through the simulation one recursive step at a time for the time span of the motion, we can establish any future state we wish to examine -- assuming we have sufficient real time and patience to do the job. Given the initial conditions, we can generate final results. To get more accurate guesses, though, we have to start with more accurate measurements of the defining properties of the system, namely the position, velocity, and mass of all of the planets.
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Stephen H Kellert gives a working definition of Chaos Theory as the specialized exploration of Dynamical Systems Theory, focusing on unstable, aperiodic behavior -- which is intrinsically unpredictable -- in deterministic nonlinear dynamical computable systems.
Kellert identifies Dynamical Systems Theory itself as the investigation of qualitative aspects of a system's behavior (as opposed to getting numerical results of equations). It may therefore be compared with Complexity Theory.
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