Beauty in the Three-Body Problem’s Chaos
Order appears to permeate the universe. From orbital resonances to the perfect spirals of galaxies, objects in the universe tend to prefer order over chaos — or at least this is how it seems.
Even if systems appear chaotic, we as scientists — astronomers, physicists, mathematicians — have proven to be quite good at understanding them thoroughly. We have studied these systems and learned how to predict their behavior. This is exactly what makes it disturbing when we can’t — when a system appears to be so chaotic that it is impossible to decipher any pattern.
The infamous three-body problem — a system of three bodies interacting gravitationally — is one of these cases; it represents one of the few areas of physics where it is taken to be true that a general, universalizable solution is impossible for us to grasp or is simply nonexistent. It is where chaos wins over order.
What is so miraculous about this case is how quickly it seems to shift from order to chaos. The two-body problem involves finding a general solution for two bodies interacting gravitationally, and we have known of this solution for centuries; we can accurately predict how two gravitating bodies will interact.
A system of three bodies simply has more unknowns than formulae with which to solve for them; we cannot find all of the terms necessary to make a general formula for the three-body problem. Except for a few special cases, no matter what we do, we always have more variables than equations to use them with.
The three-body problem has plagued scientists and mathematicians all the way back to Newton, and as such, it is a problem that has grown and evolved with history. If we build a timeline over the last three centuries, we can track the problem, understand its origins, and begin to appreciate what makes it fundamentally unsolvable. This history of the three-body problem is closely tied to the history of our understanding of gravity.
Everything began with Johannes Kepler in the early 17th century with the origin of the two-body problem — the orderly and elegantly solvable sibling of the three-body problem. Kepler is most well known for his three laws of planetary motion, published between 1609 and 1619. As their name suggests, the three laws of planetary motion were able to, for the first time, qualitatively describe the motion of planets around the Sun.
The laws are as follows:
1) Planets move around their star in elliptical orbits with the star at one focus
2) Planets cover equal areas in equal time around their orbits
3) A planet’s arbital period squared is proportional to its semi-major axis cubed
These three laws are telling us crucial information about not only how planets orbit around stars but about how any two bodies orbit around each other. The first law is telling us that all orbits are elliptical in nature; we can set a reference frame with this in mind and put the star “stationary” at one of the focuses of the ellipse — essentially reducing this two-body problem to a one-body problem.
The second law is telling us that when planets are closer to the star in their orbit, they will move faster, and when they are further, they will move more slowly — meaning that the radius vector is covering equal area across the ellipse in equal time:
And finally, the third law is telling us that the square of the period of a planet’s orbit is directly proportional to the cube of its semimajor axis — i.e., planets farther out take longer to complete their orbits.
These three laws are the seeds of what will become the more mathematically involved two and three-body problems explored by Isaac Newton. Newton expanded on the laws that Kepler proposed and eventually arrived at his universal law of gravitation; this was the first step toward solving the two-body problem.
Newton realized in 1687 that gravity obeys an inverse square law, meaning that gravity reduces proportional to the distance between the two objects squared. His solution to the two-body problem involves understanding and applying the conservation of mass, momentum, angular momentum, and energy.
While an elegant and relatively simple solution had been found for the two-body problem, the three-body problem remained unsolved. Newton began to study the motions of the Moon in relation to both the Earth and the Sun, marking the start of the three-body problem. In his Philosophiae Naturalis Principia Mathematica, Newton calls the motions of the Moon “imperfect” — in direct contrast to the perfection which we have come to expect from the physics of the universe.
Newton, in essence, introduced a problem that he couldn’t find an answer to. As Stephen Wolfram wrote in A New Kind of Science, “The three-body problem was a central topic in mathematical physics from the mid-1700s until the early 1900s.” The problem continues to evolve into the 21st century, as we begin to use computers to accurately map out more complex systems. But still, no general solution exists.
Just because the system is unpredictable from our perspective does not mean that it is random; the system is deterministic, but it is simply so sensitive to its original conditions that we have no way of universally making accurate predictions about how the system will behave. We have only been able to find a few special cases in which the orbits of three bodies are stable and don’t result in chaos or the expulsion of one of the bodies.
The mathematician Dr. Richard Montgomery gives us clear historical context: “In 1890 Henri Poincaré discovered chaotic dynamics within the three-body problem, a finding that implies we can never know all the solutions to the problem at a level of detail remotely approaching Newton’s complete solution to the two-body problem.”
This fact is disturbing to those, like myself, who believe everything must be explainable and those who love to find symmetry and order in the laws that govern the universe. It is a similar level of frustration that Einstein and many others felt about the seemingly probabilistic nature of quantum mechanics: “God does not play dice with the universe.”
We should be able to find solutions to everything; nothing is random. Yet the three-body problem remains within the domain of chaos — except for the few specific solutions we have found. But this is exactly what makes these single, special cases so interesting and extraordinary: they are specks of order within an otherwise unpredictable mathematical context.
There are five families of orbits where legitimate formulae exist within a three-body system; Leonhard Euler is credited with finding three of them, and Joseph-Louis Lagrange is credited with the two others. Euler’s first family of solutions came in 1767: the motion on collinear ellipses. It found that the three bodies can be stable and predictable so long as their positions remain collinear around the center of mass — i.e., the three bodies always form the same, straight line as they all orbit around the same point.
A Version of Euler’s Solution
Euler also gave us what is called the restricted three-body problem: a unique case in which two massive bodies are in circular orbits, and a “massless” third body is thrown into the system. This, of course, cannot formally exist in nature, but approximations can be made. We can use the restricted three-body problem if two of the bodies are so much larger than the third that the third has no noticeable gravitational effect on the other two — like in the case of a satellite in the Earth-Moon system.
Interestingly, Euler’s original solution cannot actually exist in nature either; as Dr. Juhan Frank puts it, “…these collinear solutions are not realized in nature because they are unstable to small perturbations.” The universe is full of small perturbations, so though an orderly solution has been found, we know it is ephemeral and likely only exists on physicists’ chalkboards.
Lagrange was the next in line to try at a solution. He discovered that the three-body problem could be solved by putting each of the three bodies at the vertices of an equilateral triangle. This triangle can change size and rotate, but the three bodies must remain on the vertices of the triangle and always be equidistant from each other.
A Version of Lagrange’s Solution
All three of the bodies’ orbits have the same eccentricity, but they are all oriented differently — though they all share one focus, which is their common center of mass. This can vary in form, and again, there are interesting conditions for this solution — namely, one of the three bodies must be notably larger than the other two for it to be stable.
In 2001, however, Dr. Richard Montgomery proposed a series of solutions in which there can be many bodies with the same masses; impressively, one of these solutions involves eleven bodies that create a floral orbital
arrangement. What is most remarkable about the older solutions from the 18th century, however, is that they are the only known solutions to work with arbitrary and scalable masses, so long as the ratios of the masses stay the same.
Though an infinite number of three-body problems exist, we have only been able to find solutions for a finite number of them — most of which simply do not exist in nature. It is unfortunate that this problem has dodged our attempts at a general solution for so long, but this fact only gives additional value to the beauty that we see in the special case solutions — as well as the elegance of the general solution of the two-body problem as a whole.
Of course, an infinite number of two-body problems exist as well, and miraculously, we have formulae that allow us to understand all of them. Assuming the laws of physics that we have come to know are homogeneous throughout the cosmos, we will never find a two-body problem that we do not have a solution to — and I find that to be utterly astounding.
Written by Curran Collier
2023–05–11
