The Fascinating Story Of The Three-Body Problem

A friend of mine recently wanted to hear my take on the famous three-body problem. A couple of days after sharing my views on the fascinating physics problem, I got a request from a regular reader to write about it as well. I thought to myself:

“Huh! What an interesting coincidence?!”

Well, I could not have been more wrong. It turns out that there is a new science-fiction series that goes by the same name (based on a book) that launched both of these people into investigating this problem.

Needless to say, being a science lover, I had to check the series out for myself. Although I enjoyed the work of art, I must confess that I was a tad bit disappointed that it did not go into the topic in detail.

That is where this essay comes in. My goal with this work is to present you with a beginner-friendly introduction to the three-body problem and the rich history surrounding it.

By the end of this text, you should not only be able to enjoy a conversational understanding of the topic but also appreciate the intricate, hard work put in by generations of physicists, mathematicians, and astronomers that has led us to our present understanding of the problem.

Before we get to the three-body problem though, why don’t we start with something simpler?

The One-Body Problem

Let us consider a body; more precisely, your body. Say that you are travelling along a perfectly straight, level road at a speed of 5 kilometres / Hour.

We can calculate the distance you will cover at this speed using a simple equation as follows:

Distance (in Kilometres) = Time (in Hours) * 5 (your speed)

If you keep at it for 5 hours, you will have covered 5 * 5 = 25 Kilometres. Quite simple, right?

Given the initial condition that you are travelling at a constant speed of 5 Kilometres / Hour along the road and the assumption that your speed does not vary, we can predict exactly where you will be after a given time-period.

In mathematical/physics terms, what we have here is a closed-form analytic equation with 2 known variables (your speed and the time) and 1 unknown variable (the distance you cover).

We could apply similar equations to compute the exact position of a solitary celestial body travelling through space. The problem simply changes from one dimension to three dimensions.

But what if we add a second celestial body to the mix? Cue in Johannes Keppler.

The Quest to Understand Celestial Motion

In the early 17th century, rockets and space travel were not so popular as today. Yet, people were very much interested in understanding and predicting the motion of celestial bodies they observed in the sky.

Take a wild guess as to what motivated them. Full marks to you if you guessed horoscopes, astrology, and religion — the full package!

Holy Roman emperor Rudolf II funded German mathematician and philosopher, Johannes Kepler (and his collaborators) to study the motion of celestial bodies.

Keppler achieved a great deal in his lifetime and figured out correctly that planets move in elliptical orbits around stars. Constantly faced with financial struggles and political/religious instability, he could only do so much.

But so much, he did! He came up with his own laws of planetary motion (which in dummy-terms, say the following):

1. The Law of Orbits: Planets move in elliptical orbits with the Sun at the focus.

2. The Law of Areas: Planets move faster when closer to the Sun (at perihelion) and slower when they are farther away from the Sun (at aphelion).

3. The Law of Periods: A mathematical relationship between the distance of a planet from the Sun to the time it takes to complete one full revolution around the Sun.

This inspired generations of scientists, physicists, and mathematicians, one of whom was Isaac Newton, who had already come up with Calculus and laws of motion, and was working on his law of universal gravitation.

Following the work of Kepler and his contemporaries, Newton tried to apply his knowledge to compute the long-term stability of the system of the Earth, the Sun, and the Moon.

Intriguingly enough, he noted that when there are only two bodies in space, the (gravitational) forces they exert on each other lead to a closed-form analytic equation system (just like when you were travelling along a level road). This meant that it was perfectly possible for Newton to predict their motion using mathematics and his physical laws.

However, when he added a third body (like with the Sun, the Earth, and the Moon), Newton did not end up with a solvable closed-form analytic equation system.

Instead, the system plunged into an unpredictable chaotic state. He published this problem (along with the rest of his findings) in his book Philosophiæ Naturalis Principia Mathematica.

Over the next few centuries, this so-called “three-body problem” became a fiercely competitive puzzle amongst astronomers, physicists, and mathematicians.

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The Race to Solve the Three-Body Problem

People of science are a unique bunch. But mathematicians take the cake, in my humble opinion. They are seldom excited by day-to-day applications of mathematics. But if someone says that something is impossible, they charge in droves towards pushing and prodding at the limits.

The three-body problem turned out to be such a venture for mathematicians. The rest of the world had no complaints about it either.

As science advanced, space travel became more and more viable, and solving the three-body problem showed great potential to understand our universe better. Besides that, sea navigation was also set to benefit from a better understanding of celestial motion.

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Despite the wealth of resources thrown at the problem, progress was, well, slow. In the late 19th century, genius mathematician and theoretical physicist Henri Poincaré presented his work on the topic and declared that the three-body problem CANNOT have a closed-form analytic solution.

While this sounds like a negative result, Poincaré’s work led to the birth of a field known as chaos theory, which serves as a cornerstone to many an innovation even today.

Still, none of this ever stopped mathematicians from making progress on this problem. The three-body problem is not the only problem without a closed-form analytic solution in mathematics, after all.

Typically, such a problem can be solved “approximately” using a branch of mathematics known as numerical mathematics (one of my areas of specialisation in my university days).

Think of it like this. When we try to model the whole (three-body) system using mathematics, we run into an unstable system. However, we could chop the problem into minute blocks (like meshing a geometry) in any dimension of preference.

Since Newton originally wanted to compute the “long-term” stability of the Earth, the Sun, and the Moon, it makes sense to chop the model into minute “time” blocks (meshing the temporal dimension, if you will). Now, here comes the clever bit.

None of this has changed the fact that the physical three-body problem leads to an unstable mathematical model. Instead, mathematicians approximated the problem into a comparable two-body problem, which is completely stable.

Think about it. The Moon can be considered to be in a 2-body system orbiting the Earth. The Earth and the Moon could be considered together as one mass around their center of gravity orbiting the Sun in a 2-body system. Expanding further, each planet in the solar system could be considered to be orbiting the sun in a 2-body system.

Now, we get a result for each time-step with a certain quantity of error attached to it. Over the centuries, mathematicians have been at this family of problems so hard that they have come up with ingenious ways of minimising the error and maximising the rate at which it decreases over the whole system.

Finally, we add up all of the results we have computed over the various time blocks, which is known in the biz as numerical integration. The end result was that we were able to predict the behaviour of not just three-body systems, but any arbitrary n-body system for time periods well over our lifetimes.

That’s good enough, right? Well, not quite; not for the typical mathematician, to say the least.

Mathematicians Going Above and Beyond

Our best numerical approximations are still approximations. They are bound to deviate from reality at some point. And we aren’t even considering other factors like solar-flares, hypervelocity celestial bodies making a surprise entrance into the system, etc.

This kept motivating mathematicians to continue to seek a proper solution to the three-body problem. Lo and behold, some succeeded!

In the 18th century, mathematician Leonhard Euler figured out a family of completely stable three-body configurations where three bodies orbit whilst permanently eclipsing each other (in a straight line).

Imagine the Moon and the Earth orbiting the sun, but the Moon not orbiting the Earth and the Earth permanently eclipsing the Moon.

Another famous mathematician, Joseph-Louis Lagrange figured out yet another stable three-body configuration where the three bodies form an equilateral triangle with respect to each other.

Together, these potential configurations form the Lagrange points, which the current state-of-the-art James Webb space telescope (satellite) takes advantage of. See, all that effort does amount to valuable scientific progress.

After the advent of computational power, we were able to turn this problem into a “search” problem, where we search for potential stable three-body configurations in the entire sample space. This led to further stable configurations, which helped us understand our universe further

Our present day scientists and mathematicians are still going hard at this problem and pushing the boundaries, as it helps us understand how galaxies are formed and how they evolve. Interestingly enough, most three-body configurations result in the third body being released, resulting in a stable binary pair.

If you were perceptive enough, you would have noted that all the solutions we have so far are either approximations or special case solutions (where the three bodies are in a specific geometric configuration). Does this mean that the general three-body problem still remains unsolved?

Well, the honour of solving the generalised three-body problem goes to Finnish mathematician Karl Frithiof Sundman. In the early 20th century, he came up with an infinite series that solves the problem (which is still not a closed-form solution).

The only caveat with this solution is that it converges so slow that we don’t use it for any practical calculations. Instead, we still primarily rely upon numerical approximations and special case solutions. But still, I’d say that Sundman’s solution is a net win for mathematicians.

It has been a long journey, but it has been worth it. If you have read this far, I humbly thank you for your interest, and hope that you got some insights into the lore and history surrounding the three-body problem.

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Further reading that might interest you: 

• A Tale Of Romance Between Entropy And Gravity
• 3 Reasons Why Deep Space Travel Is Really Challenging
• What Do Black Holes And The Big Bang Have In Common?

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