Long ago, there was a person for whom it is claimed that he was sitting under a specific apple tree which made him wonder why things are attracted towards the earth. Sir Issac Newton worked on his curiosity of how bodies move and which quantities play a relevant role in this motion. By lot of observations and calculations, he was able to obtain relations between how much a body moves, with what speed, how does that speed change and what time scales affect it. They are known as the equations of kinematics today. They were the first steps to understand and quantify motion of objects. These steps still remain relevant even after centuries, because of a simple reason that they work and give accurate results when it comes to you denting your father’s car or running swiftly to avoid the eyes of your professor. When Sir Newton bothered on how two individual objects interact with each other, his observations got him to the conclusion that there is a force between any two objects by the virtue of their masses. More the mass, more this force of attraction. This is the reason which makes us easier to stay attracted to the ground and makes push-ups a big challenge for a 90 kg me! This is also the reason why we do not fly away and which is also the cause of all our childhood disappointments of all the ice-creams we dropped.
The problem came when Sir Newton tried to understand how three objects interact with each other.
During the era of Newton, due to availability of telescopes and enthusiasm for observational astronomy, understanding the celestial bodies and planetary motion was an important part of science. Newton knew that the tides are due to the gravitational pull of the moon. This made understanding the motion of Sun – Moon – Earth important. But when it came to writing equations for a system of three bodies, it turned out to be a very challenging problem to solve. It is said that Newton confided to his friend Halley that the problem of Sun, Moon and the Earth “made his head ache and kept him awake so often that he would think of it no more”. This also made “Sun – Moon – Earth” as the very first properly formulated three body problem.
So, why do these three body problems pose to be complicated? What makes them so difficult to solve? The essential trick in solving a two body problem is that we can change the problem of two bodies by considering one of the bodies stationary. Two cars colliding with the same speed will yield the same damage to the cars if one of the cars was stationary and other one was moving at twice of the speed. This kind of simplification essentially helps us solve these two body problems. But when it comes to any three interacting objects, we are not able to find such a place from where this problem essentially becomes a two body problem. Why is this interesting? Because it leads to something very intriguing. Chaos.
We will discuss more about Chaos in the next edition of the article.
Nishchal Dwivedi is a Theoretical physicist at Bhabha Atomic research Centre (BARC) and a TEDx Speaker