# Three-Body Problem: Is it Solvable in Same Plane?

• Inquisiter
In summary, the three-body problem is considered unsolvable when the three bodies are not in the same plane. However, if the movement of the three bodies is coplanar, there is a solution known as the restricted three-body problem. This solution involves a series with an incredibly large number of terms to converge and is only applicable in special cases. Numerical solutions using perturbation theory have also been used to predict eclipses.
Inquisiter
Is the three-body problem still unsolvable even when the three bodies are located in the same plane?

The three body problem, in which the three bodies are not in the same plane, is so incredibly difficult that it is widely belived to have no solution.

You can't have three bodies 'not in the same plane'. They define their own plane.

rachmaninoff said:
You can't have three bodies 'not in the same plane'. They define their own plane.

but their movement need not be coplanar. the plane defined by the instantaneous position of the 3 bodies will be movin' all over the place.

r b-j

duh! I was not thinking - apologies.

-rachmaninoff

rachmaninoff said:
duh! I was not thinking - apologies.

-rachmaninoff
I wasn't thinking either actually, but yeah, what I meant is: can the 3-body problem be solved if the movement of the three bodies is coplanar? (like, when the plane defined by the three bodies isn't rotating)

You might try looking at

http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html

for some general info on the three body problem.

If you don't mind a series solution that takes 10^8000000 terms to converge :-), there is a solution to the restricted 3-body problem. The restricted three body problem is the coplanar three body problem when one of the masses is small and a circular orbit for the two "large" masses around their common COM.

The 3B problem is only solvable for some special cases.
For instance, what I think is called the "Lagrange position" with another planet directly opposite the Earth on the other side of the Sun.
Of course, numerical solution is always possible (often using perturbation theory),
and has been used for centuries to predict eclipses.

## 1. Can the Three-Body Problem be solved analytically?

The Three-Body Problem is a classic problem in physics that involves predicting the motions of three bodies that are gravitationally interacting with each other. While there are some special cases where an analytical solution has been found, in general, the Three-Body Problem cannot be solved analytically. This is due to the complex nature of the problem and the fact that it involves solving a system of non-linear differential equations.

## 2. Is the Three-Body Problem solvable in the same plane?

Yes, the Three-Body Problem can be solved in the same plane. This means that the three bodies are moving in a two-dimensional plane and their motions can be described using two-dimensional coordinates. However, it is important to note that this is a simplified version of the problem, as in reality, the three bodies may be moving in three-dimensional space.

## 3. Are numerical methods used to solve the Three-Body Problem?

Yes, numerical methods are often used to solve the Three-Body Problem. These methods involve using computers to simulate the motions of the three bodies over time. While this approach may not provide an analytical solution, it can give accurate results and allow for the exploration of different scenarios and initial conditions.

## 4. Can chaos arise in the Three-Body Problem?

Yes, chaos can arise in the Three-Body Problem. Chaos refers to the unpredictable and seemingly random behavior of a system, even though it follows deterministic rules. In the Three-Body Problem, small changes in the initial conditions can lead to drastically different outcomes, making it difficult to predict the long-term behavior of the system.

## 5. How does the Three-Body Problem relate to other areas of science?

The Three-Body Problem has connections to various areas of science, including physics, mathematics, and astronomy. It has been studied to understand the behavior of celestial bodies, such as planets and moons, in our solar system. The insights gained from studying the Three-Body Problem have also been applied in other fields, such as fluid dynamics and chaos theory.

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