Engineering is not an exact science and approximation theory lies at the very heart of it. We assume and approximate a lot, and a whole branch of mathematics called numerical analysis caters to our need. For instance, in analog electronics many parameters are inter-dependent non-linearly but we approximate and bring all that down to a neat Linear model (Remember this “Linearity …..Obvious?“). Avoiding the jargon, the output current depends on various powers of the input current in a transistor but the complexity is reduced by assuming the output current to be linearly proportional to the input (Approximating, like a Boss 🙂 ). Yeah, you can do that because the error is insignificant but, what about a system where even a small deviation can result in a chaotic ending. An analogy would be better, imagine a guy standing on the hilltop as shown. If he starts at a slightly different point, he would end up below but if he is exactly at the tip, then he stays there. So even a small variation can give rise to drastically different results. This is the premise of the big subject of 20th century, Chaos theory. Understanding Chaos theory explains how butterfly’s wings might cause tiny changes in the atmosphere which can ultimately cause a tornado on the other side of the globe. An interesting story is behind the accidental discovery of this new branch of mathematics and I am going to continue with that.
The year was 1885, King Oscar II of Sweden and Norway declared a prize of 2500 crowns to anyone who can once and for all establish, mathematically, if the solar system will keep working as it does today or will it suddenly shatter apart. The problem is as simple as it sounds? No Way. Even the great Newton, who had earlier showed that 2 bodies in space will have stable orbits, had tried and failed. The problem can be reduced to 3-body (sun, moon and the earth) but even that is a tough one; one has to deal with eighteen variables i.e. the position and velocity of each body in each of the three dimensions. The hero, also the poster boy of french mathematics then, Henri Poincaré took this challenge upon himself and started working towards this mammoth task. He invented new techniques to simplify the problem, making successive approximations to the orbits which he felt won’t affect the final outcome significantly. Eventually he couldn’t solve the problem in its entirety but he was still awarded the prize for his techniques (To all teachers and graders, do you see this? The final answer doesn’t matter, the path traversed does)
When the paper was almost ready, one of the editors realized that there was a problem. Contrary to Poincaré assumption, even a small deviation in the initial conditions can end up in completely different and unstable orbits. This observation came as a nightmare to Poincaré who had already sent a few copies of this paper out. The silver lining though was the new subject of Chaos theory which was very important and everybody soon realized its implications. In the end Poincaré became the key proponent of Chaos theory and even the short turmoil was eclipsed by the magnitude of this discovery. Maybe, Poincaré started with some good initial conditions 😉