# Mathematicians find 12,000 new solutions to 'unsolvable' 3-body problem

The three-body problem is a notoriously tricky puzzle in physics and mathematics, and an example of just how complex the natural world is. Two objects orbiting each other, like a lone planet around a star, can be described with just a line or two of mathematical equations. Add a third body, though, and the math becomes much harder. Because each object influences the others with its gravity, calculating a stable orbit where all three objects get along is a complex feat.

Now, an international team of mathematicians claims to have found 12,000 new solutions to the infamous problem — a substantial addition to the hundreds of previously known scenarios. Their work was published as a preprint to the database arXiv, meaning it has not yet undergone peer review.

More than 300 years ago, Isaac Newton wrote down his foundational laws of motion, and mathematicians have been working on solutions to the three-body problem pretty much ever since. There is no single correct answer; instead, there are many orbits that can work within the laws of physics for three orbiting objects.

Unlike our planet's simple loop around the sun, orbits for the three-body problem can look twisted and tangled, like pretzels and scribbles. The 12,000 newly discovered ones are no exception — the three hypothetical objects start at a standstill and, when released, are pulled into various spirals toward one another via gravity. They then fling past one another, moving farther away, until the attraction takes over and they once again come together, repeating this pattern over and over again.

The orbits "have a very beautiful spatial and temporal structure," lead study author Ivan Hristov, a mathematician at Sofia University in Bulgaria, told  New Scientist. Hristov and colleagues found these orbits using a supercomputer, and he's confident that with even better tech, he could find "five times more."

Three-body systems are quite common in the universe; there are plenty of star systems with multiple planets, or even multiple stars orbiting each other. In theory, these new solutions could prove extremely valuable to astronomers trying to explain the cosmos. But they're only useful if they're stable, meaning the orbital patterns can repeat over time without breaking apart, flinging one of the component worlds off into space. Just because they're theoretically stable doesn't mean they'll stand up to the many other forces present in a real star system.

"Their physical and astronomical relevance will be better known after the study of stability — it's very important," Hristov said.

Juhan Frank, an astronomer at Louisiana State University who wasn't involved in the work, is skeptical that these orbits will turn out to be stable. They're "probably never realized in nature," he told New Scientist. "After a complex and yet predictable orbital interaction, such three-body systems tend to break into a binary and an escaping third body, usually the least massive of the three."

No matter what, though, these solutions are a mathematical wonder. According to Hristov, "stable or unstable — they are of great theoretical interest."

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• Helio
I'm unclear what these 12k solutions present.

Physics, and engineering, need initial conditions. Starting with three bodies at rest seems like a non-sequitur (pun intended).

The solar system appears, so far, to have an odd orbital collection since there are more much larger planets close to their star, even after adjusting for observational bias. We seem to have the opposite. One idea is that we once had larger planets in the inner region, but they kept migrating to their demise. But that won't explain how there is so little between Earth and Jupiter, when a simple formation process from disk to planet should have produced much larger objects. Thus, the more mainstream view seems to invoke the chaos mentioned in this article, where the solar system, in its wild teenager years, became a vast pinball machine.

Also, does 12k solutions for three bodies suggest some solutions for a dozen bodies or more? We know orbital data today, so retrodictive modeling should produce some interesting results.
• billslugg
Apparently what they have found are 10,000 situations where three bodies in seemingly chaotic orbits can eventually return to the initial conditions. It is of great interest to mathematicians, but of no practical value, as best I can see.
• @whut
I'm unclear what these 12k solutions present.

Physics, and engineering, need initial conditions. Starting with three bodies at rest seems like a non-sequitur (pun intended).

The solar system appears, so far, to have an odd orbital collection since there are more much larger planets close to their star, even after adjusting for observational bias. We seem to have the opposite. One idea is that we once had larger planets in the inner region, but they kept migrating to their demise. But that won't explain how there is so little between Earth and Jupiter, when a simple formation process from disk to planet should have produced much larger objects. Thus, the more mainstream view seems to invoke the chaos mentioned in this article, where the solar system, in its wild teenager years, became a vast pinball machine.

Also, does 12k solutions for three bodies suggest some solutions for a dozen bodies or more? We know orbital data today, so retrodictive modeling should produce some interesting results.

These are natural responses. Also need to consider forced response when there may be much larger external factors. The sun may in fact be considered the eternal force.
• Nigel01
Calculating the way three things orbit each other is notoriously tricky — but a new study may reveal 12,000 new ways to make it work.

Mathematicians find 12,000 new solutions to 'unsolvable' 3-body problem : Read more
Consider a three-body star system where the three real bodies (A, B, and C) share a virtual fourth body (D) with a known position. This virtual body could be a satellite whose position remains fixed relative to the observation point. We’ll explore how this setup can help us address the three-body problem.

System Description:
Real bodies: A, B, and C (e.g., stars or planets).
Virtual body: D (the fixed-position satellite).Equations of Motion:
We’ll use Newton’s law of universal gravitation to describe the gravitational interactions between the bodies.
The gravitational force acting on each body is given by:

where:
(F_{ij}) is the gravitational force between bodies i and j.
(G) is the gravitational constant.
(m_i) and (m_j) are the masses of bodies i and j.
(r_{ij}) is the distance between bodies i and j.Equations for Each Body:
For each real body (A, B, C), we have the following equations of motion:
i}}{{dt^2}} = \sum{j \neq i} F_{ij} ]
where:
(\mathbf{r}_i) is the position vector of body i.
The sum runs over all other bodies (j ≠ i).Equation for Virtual Body D:
Since D has a fixed position, its acceleration is zero:
The gravitational force acting on D due to A, B, and C is:
The net force on D is the sum of these forces:
D = \mathbf{F}{DA} + \mathbf{F}{DB} + \mathbf{F}{DC} ]Since D’s acceleration is zero, we have:
Solving for D’s position:
Equations for Real Bodies A, B, and C:
We use the equations of motion for A, B, and C, considering the gravitational forces from D:
i}}{{dt^2}} = \sum{j \neq i} F_{ij} + \mathbf{F}_{Di} ]
where (\mathbf{F}_{Di}) is the gravitational force on body i due to D.
Numerical Integration:
Since there’s no general analytical solution for the three-body problem, numerical methods (e.g., Runge-Kutta) are used to simulate the system’s motion over time.In summary, incorporating the fixed-position virtual body D, can numerically solve the three-body problem involving A, B, and C. The equations of motion for each body, along with D’s position, allow us to track their trajectories
• COLGeek
Consider a three-body star system where the three real bodies (A, B, and C) share a virtual fourth body (D) with a known position. This virtual body could be a satellite whose position remains fixed relative to the observation point. We’ll explore how this setup can help us address the three-body problem.

System Description:
Real bodies: A, B, and C (e.g., stars or planets).
Virtual body: D (the fixed-position satellite).Equations of Motion:
We’ll use Newton’s law of universal gravitation to describe the gravitational interactions between the bodies.
The gravitational force acting on each body is given by:

where:
(F_{ij}) is the gravitational force between bodies i and j.
(G) is the gravitational constant.
(m_i) and (m_j) are the masses of bodies i and j.
(r_{ij}) is the distance between bodies i and j.Equations for Each Body:
For each real body (A, B, C), we have the following equations of motion:
i}}{{dt^2}} = \sum{j \neq i} F_{ij} ]
where:
(\mathbf{r}_i) is the position vector of body i.
The sum runs over all other bodies (j ≠ i).Equation for Virtual Body D:
Since D has a fixed position, its acceleration is zero:
The gravitational force acting on D due to A, B, and C is:
The net force on D is the sum of these forces:
D = \mathbf{F}{DA} + \mathbf{F}{DB} + \mathbf{F}{DC} ]Since D’s acceleration is zero, we have:
Solving for D’s position:
Equations for Real Bodies A, B, and C:
We use the equations of motion for A, B, and C, considering the gravitational forces from D:
i}}{{dt^2}} = \sum{j \neq i} F_{ij} + \mathbf{F}_{Di} ]
where (\mathbf{F}_{Di}) is the gravitational force on body i due to D.
Numerical Integration:
Since there’s no general analytical solution for the three-body problem, numerical methods (e.g., Runge-Kutta) are used to simulate the system’s motion over time.In summary, incorporating the fixed-position virtual body D, can numerically solve the three-body problem involving A, B, and C. The equations of motion for each body, along with D’s position, allow us to track their trajectories
Not trying to be funny, but this suggests a 4-body problem with the 4th being a fixed location.

With all calculations in reference to the fixed location, this changes the nature of the original 3-body problem.

Am I interpreting correctly?
• billslugg
In the "n-body" problem, a larger "n" would make it harder. This is a "virtual" body since the mass is insignificant. Also it does not move around. Basically just a choice of coordinates.
This is how the solutions to Einstein's field equations got upgraded several times, better choice of coordinate systems.
• COLGeek
In the "n-body" problem, a larger "n" would make it harder. This is a "virtual" body since the mass is insignificant. Also it does not move around. Basically just a choice of coordinates.
This is how the solutions to Einstein's field equations got upgraded several times, better choice of coordinate systems.
Tracking, but the fixed point changes the fundamental complexity of the n-body problem as the coordinates are in relation to that point.

Am I over simplifying?
• billslugg
Yes, in the new system each body is referenced to a fixed point in the universe. Previously two of them were referenced to the third. This made one body always at (0,0,0) I suppose, thus didn't need an equation. I guess this new system uses the barycenter as the origin since it can be fixed in space. I don't know about all this, its confusing.
• COLGeek
Yes, in the new system each body is referenced to a fixed point in the universe. Previously two of them were referenced to the third. This made one body always at (0,0,0) I suppose, thus didn't need an equation. I guess this new system uses the barycenter as the origin since it can be fixed in space. I don't know about all this, its confusing.
Seems the fixed point would allow for a "solution" for the problem that escapes original problem.

I wish I had more time to play with the math.
• Catastrophe
Physics, and engineering, need initial conditions. Starting with three bodies at rest seems like a non-sequitur (pun intended).

Exactly!..What basis is there for this assumption?

Cat :)