A person mentioning the three-body problem could be referring to a puzzle in cosmology, classical physics, or quantum mechanics. All three stem from the same difficulty, the problem of taking the starting positions and velocities of three-point masses, then figuring out their future motion using physical laws like those of Newton’s laws of motion, or theory of gravity.
Updated 25/01/2022
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This probably sounds pretty simple, just the next step in the evolution of the solvable two-body problem. But the act of adding another body to the two-body problem adds a level of complexity and makes the problem practically unsolvable.
This is because the traditional three-body problem describes a dynamical system that is chaotic for most initial conditions, and solving this requires a wealth of numerical methods.
Despite how simple a system looks, and how easy it is to describe, most are far more complex than they appear.
There is more to consider than just the bodies themselves; even in the “vacuum” of space there are gravity fields, and electromagnetic fields to consider. There are solar winds and cosmic rays. All of these compete in a cosmic tug of war of forces that adds complexity to considerations of “just three bodies.”
This makes the subjects of three-body problems almost completely unpredictable from one moment to the next, even if we know where they started.
Examples of Three-Body Problems
The classic example of a three-body problem is the system of the sun, the Earth, and the moon.
Solving this three-body problem involves calculating where the sun, moon, and Earth will all be at a certain point in time, given their initial starting positions along with their mass, current direction, and how fast they are moving.
Early navigators were the first to try to solve three-body problems, using them to steer ships across large bodies of water like the Atlantic Ocean. These efforts were prone to errors arising because of the chaotic ways that gravity exerts its influence on all three bodies when they interact.
Isaac Newton came up with some equations that involved describing the vector positions of the three bodies, assuming each had a certain mass. Solving the equations involved a long series of iterations, which is why the method was not used until computers were invented.
It is amazing to think that as much as we now know about the solar system and its worlds, there is still no general solution to the three-body problem of the Earth, its moon, and the sun.
The majority of three-body problems that researchers attempt to solve now are related to particles in systems that, while clearly much smaller, lack none of the complexity of larger systems.
A Solution to The Three-Body Problem (Sort Of…)
For a long time after the suggestion of various three-body problems, this enigma was almost filed away by physicists. A curiosity to be mulled over, but rarely seriously considered or studied.
Scientists began to once again consider searching for the solutions to such problems with the advent of powerful supercomputers capable of cracking more complex physical problems.
Part of the problem for humans solving the three-body problem is that we are intuitively very good at solving linear problems. We can take inputs and come up with outputs that are equal to them. Chaotic three-body problems are not linear in this way. The outputs do not match the inputs.
Newton’s three-body problem explained - Fabio Pacucci
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As supercomputers become more advanced, their capability to work on non-linear problems —systems in which the change of the output is not proportional to the change of the input — has burgeoned.
Yet, even for modern computers, the process of solving three-body problems is still long and laborious.
One advantage humans have over supercomputers, however, is their creativity. One way we have of solving three-body problems, sort of, is by remembering that there are solutions for the two-body problem.
So what if we made the mass of one of the bodies in the three-body problem negligible? We could just use the solution to the two-body problem to solve a reduced or “restricted” three-body problem.
In 2019, a research team from the University of Edinburgh, the University of Cambridge, Campus Universita rio de Santiago, and Leiden University wondered if neural networks could be able to succeed where humans and supercomputers have failed.
The Drunkard’s Walk Approach to The Three-Body Problem
In a new approach to the long-standing three-body problem, researchers from the Technion-Israel Institute of Technology took the random nature of the problem and turned it into a positive.
In a paper published in Physical Review X, the authors show that random and unpredictable aren’t always the same thing, using a system known as the drunkard’s walk to predict outcomes from initial conditions.
The drunkard’s walk is based upon the idea that as a drunk walks in random directions they have the same chance of taking a step to the right as taking a step to the left. If we know those chances, then it is possible to calculate the probability of the drunkard ending up in any given spot at some later point in time.
The researchers used this to assess a system in which a third body approaches the orbit of another two. The drunkard’s walk framework told them given the initial velocity of the third body the probabilities of the final state of that three-body system.
While this represents a big step towards a solution — or at least an approach to the three-body problem — there is still a long way to go before the approach can be adapted to a practical three-body system in which a multitude of forces is at play.
Solving the Three-Body Problem With Neural Networks
To find out if neural networks could solve three-body problems the team used the results of a conventional system, Brutus, built to solve the problem.
Brutus was challenged to solve 9,900 “easy” scenarios, with the data generated fed to their neural network. Then, they gave the system its own “easy” scenarios to solve and compared how it did in comparison to Brutus.
In a paper titled “Newton vs the Machine: solving the chaotic three-body problem using deep neural networks,” the team says that results from the supercomputer and the neutral network were close.
This suggested to them that a neural network is capable of solving the third-body problem. They also found it could do so more rapidly than a supercomputer, as it cracked easy problems that Brutus took minutes to solve, in just one second.
The big problem the team revealed was that without Brutus to learn from, the neural network would not have solved the easy problems at all. Despite their work being in its very early stages, they believe neural networks could eventually tackle the three-body problem.
So, has physics solved the three-body problem? Whereas we would once answer “no” we may be finally ready to replace that answer with the slightly more positive “not yet.”
References and Further Reading
Guibout. V. M., Scheeres. D. J., [2006], Modern Astrodynamics. https://experts.colorado.edu/display/pubid_284492
Sasakawa. T., Sawada. T., [1976], THREE BODY PROBLEMS WITH LOCAL POTENTIALS IN COORDINATE SPACE, Few-Body Dynamics. https://www.elsevier.com/books/few-body-dynamics/mitra/978-0-7204-0481-4
Breen. P.G., Foley. C. N., Boekholt. T., et al, [2019], ‘Newton vs the machine: solving the chaotic three-body problem using deep neural networks,’ Astrophysics of Galaxies https://arxiv.org/abs/1910.07291
Ginat. Y. B., Perets. H. B., [2021], ‘Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters,’ Physical Review X, https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031020
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