The Problem of Three Bodies
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.
Chaos in a general colloquial sense is understood as unpredictability. If you would have ever travelled (or heard stories of travelling) through Mumbai local or Delhi Metro at very crowded hours, you would surely know that when the train arrives at a platform, there is a huge gushing of passengers in and out. Now if you would be in the group of extremely experienced commuters, like me, you would know that there is a sweet spot right beside the gate of the train or metro which remains undisturbed by the motion of commuters, which we prefer to occupy. Now, if you were a newbie who is unaware and may be slightly absent minded, and you just missed that sweet spot by a few inches, the wave of commuters will hit you hard and the swarm will carry you to a very different place altogether from your initial position. In this case, a small change of the initial condition can lead to a drastic change in the final position. This is Chaos. When we are able to talk about systems which just by small changes in their initial conditions can lead to huge and unforeseen effects in final conditions, such systems are called chaotic. This is how unpredictability looks like. Real life systems like crowds, buses, career markets, populations , animal behavior , human heart , weather and even the love life of a PhD student exhibits chaos. Studying and understanding their nature means understanding such complex real life problems, and giving solutions to traffics, economics, health or may be even lesser lonely PhD students. Understanding even one of these problems gives us insight to solve other problems because they are of the same class.
So, we now understand why three body problems are interesting to study. Our interest is to see the simplest three body nucleus, the nucleus of Tritium atom (Hydrogen 3), or Triton. Hydrogen atom just has one proton in the nucleus and one electron orbiting around it. Triton has an additional of two neutrons sitting in the nucleus. Since the charge of the nucleus is given only by the proton and the neutrons are electrically neutral, the neutrons just increase the mass of this hydrogen nucleus. Such forms of the same atom having different masses (and different number of neutrons) are called as isotopes. This gives us a single proton and two neutrons sitting closely together and bound inside this nucleus, making it a three body problem.
What makes it more interesting is something to do with the small size of this nucleus. To get a perspective of the size of this object, if we take one meter and slice it in Quadrillion (one followed by 15 zeros) equal parts, one of the part will be the size of this object called as Triton. At such small scales of length, we observe something very interesting. Any particle-like object of such small scale, exhibits significant wave nature. What it means is that if we take a collection of such particles and pass them through a sieve which has only two slits, instead of piling up in front of these two slits like particles, they will show beautiful alternating patterns of varying intensity, just as if a water wave hit those two slits and both the slits became individual source of another wave coming out from them and they merged to form a pattern. Such small particles follow something called as Quantum Mechanics.
Understanding and solving this quantum three body problem is essential. Our sun continuously provides us with energy using nuclear fusion process. If we are able to recreate this fusion process successfully on earth, we will be able to solve the energy problems of our planet, providing us with a great source of energy which does not pollute. The most important part of this fusion nuclear reaction happening in the sun is the fusion of the Deuteron (the nucleus of Hydrogen-2) and Triton (the nucleus of Hydrogen-3) into one, which leads to release of energy. The Deuteron is a well understood two problem. Understanding Triton will make us understand how these fusions occur and help us calculate necessary quantities which we need for creating a fusion reactor on earth, hence contributing towards the solution of earth’s energy problem.
To solve this problem, we need to invoke certain symmetries and mathematics specific to the nucleus and we have been able to solve this. But how do we know what we did is correct? Here comes the power of experiments. There have been numerous experimental observations which are able to give us some numbers of certain quantities which the experimenters can measure. Now, from our calculations and equations, we are able to calculate those quantities which are experimentally observed. If they match, then we can say that our theory and all the calculations we did are correct. In our case, this treatment yields us the experimentally correct result for the observed components of the wave function of Triton. We are able to solve the simplest nuclear quantum three body problem.
Understanding physical systems with theoretical physics involves venturing through a lot of mathematics and imagination. Many a times to an untrained eye, this mathematics seems to be unnecessary to learn and quiet hypothetical. But once we train ourselves enough, we will be able to see that this math which we may call fancy is actually the way nature speaks to us. It is the language waiting to be decoded and waiting to be put together by our wit for understanding a specific problems we would like to study (like Triton). If we stay unaware of this knowledge and someday nature tries to speak with us, we might not be able to listen.