# How Was Chaos Theory Discovered?

*Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.*

Chaos is a term with different meanings for different people. Some use it to identify how their lives work; others use it to describe their art or the work of others. For scientists and mathematicians, chaos instead can talk about entropy of the seemingly infinite divergences we find in physical systems. This chaos theory is predominant in many fields of study, but when did people first develop it as a serious branch for research?

## Physics is Nearly Solved…Then Not

To fully appreciate the rise of chaos theory, know this: by the early 1800s, scientists were sure that determinism, or that I can determine any event based off a prior one, was well accepted as fact. But one field of study escaped this, though it did not deter scientists. Any many-body problem like gas particles or solar system dynamics were tough and seemed to escape any easy mathematical model. After all, interactions and influences from one thing to another are *really* hard to solve because conditions are constantly changing (Parker 41-2)

Fortunately, statistics exists and was used as an approach to solve this conundrum, and the first major update on the theory of gas was done by Maxwell. Prior to them, the best theory was by Bernoulli in the 18^{th} century, in which elastic particles hit each other and thus cause pressure on an object. But in 1860 Maxwell, who helped develop the field of entropy independent of Boltzmann, found that Saturn’s rings had to be particles and decided to use Bernoulli’s work on gas particles to see what could be wrought from them. When Maxwell plotted the velocity of the particles, he found that a bell shape appeared – a Normal distribution. This was *very* interesting, because it seemed to show that a pattern was present for a seemingly random phenomenon. Was there something more going on? (43-4, 46)

Astronomy always begged that very question. The heavens are vast and mysterious, and understanding the properties of the Universe was paramount for many scientists. Planetary rings were definitely a big mystery, but more so was the Three Body Problem. Newton’s laws of gravity are very easy to calculate for two objects, but the Universe isn’t so simple. Finding a way to relate the motion of three celestial objects was very important as to the stability of the solar system…but the goal was challenging. The distances and influences of each on the others was a complex system of mathematical equations, and a total of 9 integrals cropped up, with many hoping for an algebraic approach instead. In 1892, H. Bruns showed that not only was that impossible, but that differential equations were going to be key to solving the Three Body Problem. Nothing involving momentum nor position was conserved in these problems, attributes that many introductory physics students will attest is the key to solvability. So how does one proceed from here (Parker 48-9, Mainieri)

One approach to the Problem was to start with assumptions and then get more generic from there. Imagine that we have system where the orbits are periodic. With the correct initial conditions, we can find a way to get the objects to eventually return to their original positions. From there, more details could be added on until one could arrive at the generic solution. Perturbation theory is key to this building up process. Over the years, scientists went with this idea and did get better and better models…but no set mathematical equation that didn’t require some approximations (Parker 49-50).

## Stability

The gas theory and Three Body Problem both hinted at something missing. They even implied that math might not be able to find a stable state. This then leads one to wonder if any such system is stable *ever*. Does any change to a system cause a total collapse as changes spawn changes which spawn changes? If the summation of such changes converged, that implies that the system will eventually stabilize. Henry Poincare, the great mathematician of the late 19^{th} and early 20^{th} century decided to explore the topic after Oscar II, the king of Norway, offered a cash prize for the solution. But at the time, with over 50 known significant objects to include in the solar system, the stability issue was tough to pinpoint. But undeterred was Poincare, and so he started with the Three Body Problem. But his approach was unique (Parker 51-4, Mainieri).

The technique utilized was geometric and involved a graphing method known as phase space, which records position and velocity as opposed to the traditional position and time. But why? We care more about how the object is moving, the dynamics of it, rather than the time frame, for the motion itself is what lends to stability. By plotting how objects are moving in phase space, one can then extrapolate its behavior overall, usually as a differential equation (which are just so lovely to solve). By seeing the graph, solutions to the equations can become clearer to see (Parker 55, 59-60).

And so for Poincare he used phase space to create phase diagrams of Poincare sections, which were little sections of an orbit, and recorded the behavior as the orbits progressed. He then introduced the third body, but made it much less massive than the two other bodies. And after 200 pages of work, Poincare found…no convergence. No stability was seen or found. But Poincare still got the prize for the effort he expended. But before he published his results, Poincare reviewed the work carefully, to see if he could generalize his results. He experimented with different setups and found that patterns were indeed emerging, but of divergence! Now totaling 270 pages, the documents were the first hints of chaos in the solar system (Parker 55-7, Mainieri).

## Works Cited

Mainieri, R. “A brief history of chaos.” *Gatech.edu*.

Parker, Barry. Chaos in the Cosmos. Plenum Press, New York. 1996. Print. 41-4, 46, 48-57.

**© 2018 Leonard Kelley**