Three Body Problem

The Three Body Problem is a four hundred year old problem of mathematics which has its roots in the unsuccessful attempts to simulate a heliocentric Sun-Earth-Moon system.

Due to the nature of Newtonian Gravity, a three body system inherently prefers to be a two body orbit and will attempt to kick out the smallest body from the system—often causing the system to be destroyed altogether. There are a limited range of scenarios in which three body orbits may exist. It is seen that those configurations require at least two of the three bodies to be of the same mass, can only exist with specific magnitudes in specific, sensitive, and highly symmetrical configurations, and exhibit odd loopy orbits that look quite different than the systems of astronomy proposed by Copernicus. The slightest imperfection, such as with bodies of different masses, or the effect of a gravitational influence external to the system, causes a chain reaction of random chaos which compels the entire system to fall apart.

—Paul Trow, Chaos and the Solar System

400 Years of Defiance
In The Physics Problem that Isaac Newton Couldn't Solve (Archive) physics and astronomy professor Robert Scherrer (bio) tells us:

Newton's Solution
Mathematician and astronomer Issac Newton is credited to have "brought the laws of physics to the solar system."1 To solve the multi-body problems of his system Newton famously invoked divine intervention (Archive):

1The University of California San Diego credits Newton with providing the laws of physics for the Solar System (Archive):

Henri Poincaré
http://n.ethz.ch/~stiegerc/HS09/Mechanik/Unterlagen/Lecture13.pdf (Archive)

Ask a Mathematician
https://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/ (Archive)

No General Solutions
Physics Professor Richard Fitzpatrick at the University of Texas says (Archive):

Science China Press says (Archive):

From The Three-Body Problem (Archive) by Z.E. Musielak and B. Quarles we see:

As we read above, there are no general solutions to the Three Body Problem and that the motions are unpredictable. As the authors of "Ask A Mathematician" related, the only solutions require very specific and odd scenarios.

A Thousand New Solutions
In 2017 researchers used a supercomputer to test various configurations and reported over one thousand new special solutions to the Three Body Problem. We read an account from a New Scientist article titled Infamous three-body problem has over a thousand new solutions (Archive) :

As suggested by the above article, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."

Two Bodies of Equal Mass
The research group with the super computer published another study titled: Over a thousand new periodic orbits of a planar three-body system with unequal masses (Archive). The group found only special solutions; configurations where two bodies had the same mass.

Hill's Region
Astronomer and mathematician George William Hill studied the Three Body Problem. The only way Hill was able to make any progress at all was by using the Restricted Three Body Problem, where one of the bodies was of zero or negligible mass. Even then, the body was still chaotic. The benefit of the Restricted Three Body Problem and the Mass-less moon meant that that the moon would be no longer ejected from the system, as it would usually be. It is confined to what is known as "Hill's Region".

From http://www.scholarpedia.org/article/Three_body_problem (Archive):



The above depicts a crazy and chaotic moon which even makes a u-turn in mid orbit.

From the text that accompanies the image:

"Zero mass body" -- One of the bodies in the restricted three body problem is of zero mass.

"Nevertheless this does not prevent collisions with the earth" -- It's still chaotic, even in that simplified version.

One might observe that Newtonian Mechanics most certainly does not naturally default to the heliocentric system of Copernicus.

Poliastro
Poliastro, an astrodynamics software developer, shares several numerical methods for the restricted three body problem:

https://twitter.com/poliastro_py/status/993418078036873216?lang=en (Archive)



Arenstorf orbits are the knotted closed trajectories which are the result of Restricted Three Body Problem solutions.

Chaos Theory: A Demo
The available solutions to the Two and Three Body Problems are so sensitive that the slightest change or imperfection will tear the entire system apart. From 'Mathematics Applied to Deterministic Problems in Natural Sciences' we read another account of Poincaré's discoveries:

The orbits are incredibly sensitive. As a very illustrative demonstration, take a look at this online Three Body Problem simulator that uses the simplest possible figure eight pattern, which requires three identical bodies of equal mass which move at very specific momentum and distance in relation to each other.

Demo: Figure-Eight Three Body Problem



Adjust the slider values in the upper left of the simulation to something very slight to find what happens. What you will see is a demonstration of Chaos Theory. Any slight modification to a perfect system creates a chain reaction of random chaos.

This is precisely the issue of modeling the Heliocentric System, and why its fundamental system cannot exist. Only very specific, sensitive, and highly symmetrical configurations may exist. The slightest deviation, such as with a system with unequal masses, or the minute influence from a gravitating body external to the system, will cause the entire system to fly apart. The reader is invited to decide for his or her own self whether those scenarios would occur in nature as described by popular theory.

N-Body Solution Galleries
Below are links to galleries of various Three Body and n-Body solutions which have been found. One should note the lack of heliocentric orbits and mass restrictions.

Wolfram Science

https://www.wolframscience.com/nks/notes-7-4--three-body-problem/ (Archive)



Institute of Physics Belgrade

http://three-body.ipb.ac.rs/ (Archive)

n-Body Choreographies

http://rectangleworld.com/demos/nBody/ (Archive) - Notice that the common theme is highly symmetrical shapes.

Scholarpedia.org

http://www.scholarpedia.org/article/Three_body_problem (Archive)

Euler Math Toolbox

http://euler.rene-grothmann.de/Programs/Examples/Three-Body%20Problem.html (Archive)

Supercomputing Challenge
Programming students participated in the New Mexico Supercomputing Challenge to simulate the solar system and found issues with creating basic orbits:

Simulation of Planetary Bodies in the Universe (N-Body) (Archive) (Source Code)

The PDF report explains with additional details that all attempts at observing orbit were unsuccessful.

Universe Sandbox 2
Universe Sandbox 2 (Universe Sandbox ²) is a physics based space simulation developed and published by Giant Army. It has often been claimed that this simulation provides evidence that the Sun-Earth-Moon System and the Solar System are able to be simulated with gravity.

Read the following from a developer blog post and decide whether the program is using a full simulation of gravity:

Working Through the N-Body Problem in Universe Sandbox ² (Archive)

One might ponder that, surely, if all they had to do was to wait days for the simulation to advance millions of years for the 'sun to evolve' in order for heliocentric orbits to become possible, they could simply run that simulation and save its state to use as a baseline for further simulations.

Despite any fanciful allusions that the choice is between either "destabilizing the orbits with massive errors" or "waiting for Sun to evolve," and reassurances that our solar system is stable despite the inability of simulating it—all common statements to excuse away the problems of simulating bodies in heliocentric orbits—its developers freely admit that the program is not using a full simulation of gravity. The article states "By default, the simulations in Universe Sandbox ² try to set an accuracy which prevents orbits from falling apart due to error".

Sean Carrol
Caltech physicist Sean Carrol (bio) restates the same as all of the above on his page N-Bodies (Archive), starting with the history of the problem.

Sean Carrol describes the three body problem orbits as chaotic and classifies the special orbits which were discovered as "highly-symmetric." Dr. Carrol proceeds to give animations of figure eight configurations and other special symmetrical orbits which have been discovered.



Readers should decide for themselves whether the Sun-Earth-Moon system or other systems proposed by contemporary Astronomy are "highly symmetric" systems. Recall, also, what happened to the interactive figure eight demo when a slight modification to the masses or velocity was made: The slightest imperfection to the perfect symmetrical system caused a chain reaction of chaos which caused the entire system to fly apart.

Analytical Vs. Numerical
Q. I think those quotes are talking about analytical solutions. There are working numerical solutions...

A. This is a misconception which stems from some sources which state that there are no analytical solutions, only numerical solutions. This might cause a casual reader to assume that there must be solutions in which the conventional systems of astronomy work. While it is true that the analytical approach of creating an equation to predict future positions based on initial conditions is much more difficult, the working 'numerical solutions' are the special cases described above -- the figure eight and other highly symmetric configurations.

The "numerical solutions" require at least two of the three bodies to be of the same mass.

Over a Thousand New Solutions - New Scientist
From the New Scientist article Infamous three-body problem has over a thousand new solutions we read:

Clicking on the arxiv.org source at the bottom of the that article takes us to the paper The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum, where we see at the bottom of p.1:

Elsewhere in the paper it describes:

Over 600 New Orbits
Similarly, the phys.org article Scientists discover more than 600 new periodic orbits of the famous three-body problem (Archive) describes the discovery of other symmetrical orbits:

Figure Eight
The famous symmetrical Figure Eight problem was discovered numerically:

http://numericaltank.sjtu.edu.cn/three-body/three-body.htm (Archive)

1349 New Families
Over a thousand new periodic orbits of a planar three-body system with unequal masses

Further down in the same paper, in the section "Numerical searching for periodic orbits" we verify that these are numerical simulations:

We see that these special solutions are the numerical solutions. Just where are the solutions with different masses? Opponents are unable to show that there are solutions with different masses, that there are non-symmetrical configurations, or that the Sun-Earth-Moon system can be simulated by the Three Body Problem.

Vinay Ambegaokar
In the Chaos chapter (Archive) of Reasoning about Luck: Probability and Its Uses in Physics, its author Physics professor Vinay Ambegaokar (bio) tells us that the three-body problem is not studied in the physics curriculum:

Erwin B. Montgomery
On p.174 of Medical Reasoning: The Nature and Use of Medical Knowledge (Archive) its author Dr. Erwin B. Montgomery shares this quote from Newton:

Robert Rosen
See the following quote from Fundamentals of Measurement and Representation of Natural Systems by Professor Robert Rosen (bio):

The above states that the due to the issues of the three body problem we cannot answer questions such as whether the earth-sun-moon system is stable or not. This suggests that it cannot be simulated.

Mark Cunningham
On p.189 of Neoclassical Physics by theoretical physicist Mark Cunningham Ph.D. (bio) we read the following (Archive):

Classical Mechanics

Yet again, we find direct statements telling us that the venture of describing or simulating the earth-sun-moon system has been unsuccessful, from the time of Newton up to present. Numerous and notable mathematicians over the years have attempted to describe or simulate the earth-sun-moon system without success—telling us everything that we need to know about the systems of Newton and Copernicus.

Stability of the Solar System
It has been asserted that Laplace and Lagrange demonstrated the stability of the Solar System. On this topic, Professor H. Scott Dumass tells us in The KAM Story (Archive):