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

[Admin]

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

 

[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? [rhetorical] 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? [rhetorical] 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.

1. System Description:
   • Real bodies: A, B, and C (e.g., stars or planets).
   • Virtual body: D (the fixed-position satellite).
2. 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:
   • [ F_{ij} = G \frac{{m_i m_j}}{{r_{ij}^2}} ]
   
   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.
3. Equations for Each Body:
   • For each real body (A, B, C), we have the following equations of motion:
   • [ m_i \frac{{d^2 \mathbf{r}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).
4. Equation for Virtual Body D:
   • Since D has a fixed position, its acceleration is zero:
   • [ m_D \frac{{d^2 \mathbf{r}_D}}{{dt^2}} = 0 ]
   • The gravitational force acting on D due to A, B, and C is:
   • [ F_{DA} = G \frac{{m_D m_A}}{{r_{DA}^2}} ] [ F_{DB} = G \frac{{m_D m_B}}{{r_{DB}^2}} ] [ F_{DC} = G \frac{{m_D m_C}}{{r_{DC}^2}} ]
   • The net force on D is the sum of these forces:
   • [ \mathbf{F}D = \mathbf{F}{DA} + \mathbf{F}{DB} + \mathbf{F}{DC} ]
   • Since D’s acceleration is zero, we have:
   • [ \mathbf{F}_D = m_D \mathbf{a}_D = 0 ]
   • Solving for D’s position:
   • [ \mathbf{r}_D = \frac{{m_A \mathbf{r}_A + m_B \mathbf{r}_B + m_C \mathbf{r}_C}}{{m_A + m_B + m_C}} ]
5. 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:
   • [ m_i \frac{{d^2 \mathbf{r}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.
6. 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.

1. System Description:
   • Real bodies: A, B, and C (e.g., stars or planets).
   • Virtual body: D (the fixed-position satellite).
2. 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:
   • [ F_{ij} = G \frac{{m_i m_j}}{{r_{ij}^2}} ]
   
   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.
3. Equations for Each Body:
   • For each real body (A, B, C), we have the following equations of motion:
   • [ m_i \frac{{d^2 \mathbf{r}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).
4. Equation for Virtual Body D:
   • Since D has a fixed position, its acceleration is zero:
   • [ m_D \frac{{d^2 \mathbf{r}_D}}{{dt^2}} = 0 ]
   • The gravitational force acting on D due to A, B, and C is:
   • [ F_{DA} = G \frac{{m_D m_A}}{{r_{DA}^2}} ] [ F_{DB} = G \frac{{m_D m_B}}{{r_{DB}^2}} ] [ F_{DC} = G \frac{{m_D m_C}}{{r_{DC}^2}} ]
   • The net force on D is the sum of these forces:
   • [ \mathbf{F}D = \mathbf{F}{DA} + \mathbf{F}{DB} + \mathbf{F}{DC} ]
   • Since D’s acceleration is zero, we have:
   • [ \mathbf{F}_D = m_D \mathbf{a}_D = 0 ]
   • Solving for D’s position:
   • [ \mathbf{r}_D = \frac{{m_A \mathbf{r}_A + m_B \mathbf{r}_B + m_C \mathbf{r}_C}}{{m_A + m_B + m_C}} ]
5. 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:
   • [ m_i \frac{{d^2 \mathbf{r}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.
6. 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