THE PERPLEXING THREE-BODY PROBLEM
Henri Poincaré
Near the end of the 19th century, the King of Sweden offered a substantial cash prize to anyone who could solve the “n-body” problem. This problem dealt with the motion of a number of bodies which do not collide but whose gravitational forces act on each other. The purpose was to investigate the stability of our solar system.
The French mathematician Henri Poincaré knew that the “n-body” problem was extremely difficult. Consequently, he attempted to start by solving the (what he thought should be) simpler “three-body problem.”
Simply stated, Poincaré’s first goal was to determine the future motion (position and velocity) of three bodies, such as three somewhat closely positioned stars or planets of comparable mass, which exert gravitational forces on each other and never collide.
The initial position and initial velocity of each body were assumed to be known. These are called the “initial conditions” for the problem.
In principle, the problem was easy. Ob-taining the solution to even this “simpler” problem, however, had perplexed scientists and mathematicians for years.
Newton’s law of gravitation, together with his laws of motion, clearly indicated how the problem should be formulated. Newton, himself, had solved the considerably easier two-body problem some two hundred years earlier.
Solving the two-body problem enabled people to accurately predict the motion of a planet around the sun or of the moon around the earth. (The influences of all the other many celestial bodies were ignored.)
Despite the fact that Poincaré was an extremely brilliant mathematician, he was not able to solve even the three-body problem completely. (Neither was anyone else at the time.)
The gravitational force of one body on another varies inversely as the SQUARE of the distance between them. Thus, simultaneous nonlinear differential equations result and these are very difficult to solve using traditional mathematical techniques. Computers, which now make such problems readily solvable, did not exist at the time.
POINCARÈ’S IMPORTANT DISCOVERY
Poincaré, however, did make a very important observation. He found that very small differences in where the bodies start produce very large differences in where they are found later.
Although the significance of this finding was not recognized at the time, it eventually would be determined to be a crucial element in establishing the existence of chaos. The phenomenon came to be known as “extreme sensitivity to initial conditions.”
More than sixty years would elapse before the phenomenon of chaos would be named and its importance would be recognized. Until then, it was merely considered to be an “unexplained phenomenon” or a “nuisance” associated with certain problems.
(This description of Poincaré’s work was summarized from “Engines of Our Ingenuity, No. 2598, The Three Body Problem” by Andrew Boyd on the University of Houston website.) A copy of that paper can be read by clicking on the following:
Alternatively, you can click on the blue link below to read this paper and other interesting papers in the “Engines of Our Ingenuity” series on the UH website:
LORENZ AND THE
BUTTERFLY EFFECT
Edward Lorenz
Edward Lorenz was doing meteorology research at MIT in the late 1950’s and was interested in seeing if weather predictions could be made by modeling the atmosphere using 12 simultaneous nonlinear fluid flow equations. He solved the equations using a primitive (by today’s standards) electronic computer that provided results with a precision of six decimal places.
One day, Lorenz decided to rerun a portion of his data in order to examine the results in greater detail. To save time, he entered the initial conditions from the results generated by a previous run, but to a precision of only three decimal places. He anticipated that this would cause negligible error and that the new results would follow the original data.
Lorenz restarted the computer and went for a cup of coffee. Upon returning about an hour later, he was amazed to find that the new results at first had followed those generated previously but soon began diverging by an ever increasingly greater amount. Lorenz soon determined that the cause was the fact that he had rounded off the original data when restarting the computer run.
Clearly, Ed Lorenz had observed the phenomenon of “extreme sensitivity to initial conditions” that Poincarè first encountered many years earlier. Lorenz quickly realized that, as a result of this extreme sensitivity, long-range weather forecasting would never be possible.
The extreme sensitivity to initial conditions phenomenon in weather prediction eventually became more simply known as the “butterfly effect.” It implied that extremely small changes in weather conditions in one part of the world (the flapping of a butterfly’s wings, for example) might, in theory, produce significant weather changes (a tornado or a hurricane, perhaps) in another part of the world.
At first, few knew about or paid much attention to the significance of Lorenz’ work. Before long, however, researchers in a wide range of other disciplines began to wonder if some of the inexplicable anomalies in their own calculations might be due to the same extreme sensitivity to initial conditions phenomenon.
Fast digital computers were becoming more widespread. Much more careful analyses of the effects of initial conditions were now possible and were begun.
The study to understand these effects soon began in earnest. It should be noted, however, that the now familiar term “chaos” was not coined until 1975 in a paper by Tien-Yien Li and James A. Yorke.
Click on the following link to read a more complete description of Lorenz’ work. This article was published in the MIT Technology Review.
If you prefer to go to the MIT Technology Review website to read this and other articles, you can click on the following blue link:
Clicking on the “LORENZ TRIBUTE” link below will let you read a great tribute to Edward Lorenz written by Professor J. C. Sprott of the University of Wisconsin at Madison.
This paper was found at Professor Sprott’s outstanding website which has material on chaos, fractals and a variety of other interesting topics. You can reach his website by clicking on the blue link below.
Our discussion of Lorenz’ work continues on page 4 of this website. Be sure and read it.
