Newton, Determinism, and Chaos


David Atkatz, Department of Physics

     ...in November [I] had the direct method of fluxions...in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon...& thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age of invention & minded Mathematicks & Philosophy more than at any time since.    

Sir Isaac Newton

    So an aged Isaac Newton laconically related how he had, as a 23-year old graduate of Trinity College, Cambridge, forever changed our view of our universe. Describing here his invention of the differential and integral calculus, and his discovery of the law of gravitation, he had been a clever fellow indeed. His above-mentioned proof that gravity, the force responsible for an apple's fall, holds the moon in her orbit as well, united in a single stroke heaven and earth, and showed that both obey a single set of universal laws. He elucidated the laws of motion, and humankind, for the first time, was given precise predictive power over nature. Newton's laws, and his calculus, allowed the future to be foretold.

    How, exactly? Insert into Newton's "equation of motion" an object's initial position and velocity, and a description of all the forces acting upon it, and turn the crank--calculate yourself, or have a computer do it for you--and out comes that body's position and velocity at all later times. This is true, too, for collections of objects: given the initial conditions--the position and velocity of each body--and the acting forces, the entire future history of that system is determined uniquely. Thus, at the end of the eighteenth century (just about the time of Thomasina's imagined birth), the French mathematician Pierre Simon de Laplace could describe "determinism":

    If an intelligence, for one given instant, recognizes all the forces which animate Nature, and the respective positions of the things which compose it, and if that intelligence is sufficiently vast to subject these data to analysis, it will comprehend in one formula the movements of the largest bodies of the universe as well as those of the minutest atom: nothing will be uncertain to it, and the future as well as the past will be present to its vision.     

    (For "intelligence," we moderns may read "computer." Computers do these kind of calculations wonderfully well; it's what they live for.) Repeat an experiment--start a system off with the same initial conditions, and the same motion results each time, exactly as Newton predicts.

    Newton's mechanics are most easily applied to systems whose motions are regular--planets revolving around suns, or ripples on a pond. We can calculate, for example, the solar eclipse schedule centuries into the future. Our understanding of the laws of motion enable us to do wondrous things--send a single spacecraft on a "grand tour" of the outer planets and their moons. But things are never as simple as we'd like for them to be. As it happens, determinism--the certainty of Newton's laws--does not always imply predictability. Some physical systems seem to defy our ability to predict. Repeat an experiment with identical initial conditions, and very soon the body's position and velocity do not even come close to those obtained the first time. After a very short time, the state of the system will be essentially unknown. These systems, which can be as "simple" as water dripping from a faucet, are called "chaotic." It is virtually impossible to predict when the next drop will fall. What's happening?

    Another French mathematician, Henri Poincare, was the first to realize that the problem lay not with the laws of motion, for those are truly universal, but with the specification of the initial conditions.

    ...it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error [change] in the former will produce an enormous error [change] in the latter. Prediction becomes impossible...    

    Chaotic systems, while obeying Newton's laws, exhibit this "sensitive dependence on initial conditions." It is not our ability to calculate that does us in, it is our inability to measure. To repeat an experiment, we must start the system off with precisely the same initial conditions. But we cannot measure that precisely--the system starts slightly differently each time, and that unknown difference is rapidly magnified, so that very soon we no longer have any idea where each body will be. Deterministic laws, subject to uncertain data, have led to complete ignorance.

David Atkatz, an Associate Professor of Physics at Skidmore, was born and raised in the Bronx. He received a Ph.D. in Theoretical Physics from SUNY at Stony Brook in 1979, and, one way or another, has been doing physics ever since.