Chaos - James Gleick [19]
Small nonlinearities were easy to disregard. People who conduct experiments learn quickly that they live in an imperfect world. In the centuries since Galileo and Newton, the search for regularity in experiment has been fundamental. Any experimentalist looks for quantities that remain the same, or quantities that are zero. But that means disregarding bits of messiness that interfere with a neat picture. If a chemist finds two substances in a constant proportion of 2.001 one day, and 2.003 the next day, and 1.998 the day after, he would be a fool not to look for a theory that would explain a perfect two-to–one ratio.
To get his neat results, Galileo also had to disregard nonlinearities that he knew of: friction and air resistance. Air resistance is a notorious experimental nuisance, a complication that had to be stripped away to reach the essence of the new science of mechanics. Does a feather fall as rapidly as a stone? All experience with falling objects says no. The story of Galileo dropping balls off the tower of Pisa, as a piece of myth, is a story about changing intuitions by inventing an ideal scientific world where regularities can be separated from the disorder of experience.
To separate the effects of gravity on a given mass from the effects of air resistance was a brilliant intellectual achievement. It allowed Galileo to close in on the essence of inertia and momentum. Still, in the real world, pendulums eventually do exactly what Aristotle’s quaint paradigm predicted. They stop.
In laying the groundwork for the next paradigm shift, physicists began to face up to what many believed was a deficiency in their education about simple systems like the pendulum. By our century, dissipative processes like friction were recognized, and students learned to include them in equations. Students also learned that nonlinear systems were usually unsolvable, which was true, and that they tended to be exceptions—which was not true. Classical mechanics described the behavior of whole classes of moving objects, pendulums and double pendulums, coiled springs and bent rods, plucked strings and bowed strings. The mathematics applied to fluid systems and to electrical systems. But almost no one in the classical era suspected the chaos that could lurk in dynamical systems if nonlinearity was given its due.
A physicist could not truly understand turbulence or complexity unless he understood pendulums—and understood them in a way that was impossible in the first half of the twentieth century. As chaos began to unite the study of different systems, pendulum dynamics broadened to cover high technologies from lasers to superconducting Josephson junctions. Some chemical reactions displayed pendulum-like behavior, as did the beating heart. The unexpected possibilities extended, one physicist wrote, to “physiological and psychiatric medicine, economic forecasting, and perhaps the evolution of society.”
Consider a playground swing. The swing accelerates on its way down, decelerates on its way up, all the while losing a bit of speed to friction. It gets a regular push—say, from some clockwork machine. All our intuition tells us that, no matter where the swing might start, the motion will eventually settle down to a regular back and forth pattern, with the swing coming to the same height each time. That can happen. Yet, odd as it seems, the motion can also turn erratic, first high, then low, never settling down to a steady state and never exactly repeating a pattern of swings that came before.
The surprising, erratic behavior comes from a nonlinear twist in the flow of energy in and out of this simple oscillator. The swing is damped and it is driven: damped because friction is trying to bring it to a halt, driven because it is getting a