# Has the Three Body Problem Ever Been Solved?

• tim_lou
In summary: Sundman's solution does exist, it may not be practical or even possible to use.In summary, the three body problem has not been solved analytically, but there are many solutions for specific configurations.
tim_lou
is the three body problem ever solved? i heard some guys in MIT solved it couple years ago... but I do not know if it's true...

I heard that Sundman has a complete solution to the 3 problem using convergent series... even though it converges soooo slowly...

Can someone post the equation Sundman came up with? (even though i may not really understand it...) or informatios about the solution to the three body problem?

As far as I am aware, there is no analytic solution to the general 3 body problem. However, a number of solutions exist for certain specific configurations of the bodies.

I haven't heard of Sundman, so can't comment on his solution.

I know that you can solve the problem for specific situations (1 body fixed, two bodies fixed, three bodies with the same mass, etc.) but essentially what has to happen is that you have to make some sort of requirement that reduces the number of degrees of freedom of the system in order to really write down any kind of meaningful differential equation that you can outright solve.

what about the restricted three body problem, where two of the bodies are un-influenced by the third body?

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Excuse my ignorance...
Can someone please post a link to the 3 body problem or something... I am very interested...
Thx

tim_lou said:
what about the restricted three body problem, where two of the bodies are un-influenced by the third body?

Then this is trivially a two body problem with a third body just passing by. You have to have some kind of interaction or else the problem separates.

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

and

http://en.wikipedia.org/wiki/N-body_problem

has some pretty good general information on the status of the three body problem.

Note that numerical integration of the differential equations is well-known. Every once in a while someone gets confused over the difference between our ability to numerically integrate from initial conditions and our inability to write down closed form algebraic solutions.

The wikipedia article explains it best:

n the physical literature about the n-body n >= 3 sometimes the statement can be found about the impossibility of solving the n-body problem. (This seems to be similar to theorems by Abel and Galois about the impossibility of solving algebraic equations of degree higher than five by means of formulas only involving roots). However one has to be careful here. This statement is based on the method of first integrals.

The n-body problem contains 6n variables, since each point particle is represented by 3 space and 3 velocity components. First integrals (for ordinary differential equations) are functions that remain constant along any given solution of the system, the constant depending on the solution. In other words, integrals provide relations between the variables of the system, so each scalar integral would normally allow the reduction of the system's dimension by one unit. Of course, this reduction can take place only if the integral is an algebraic function not very complicated with respect to its variables. If the integral is transcendent the reduction can not be performed.

The n-body problem has 10 independent algebraic integrals

1. 3 for the centre of mass
2. 3 for the linear momentum
3. 3 for the angular momentum
4. 1 for the energy.

This allows the reduction of variables to 6n-10 . The question at that time was whether there exists other integrals besides these 10. The answer was given in 1887 by H. Bruns.

Theorem (First integrals of the n-body problem) The only linearly independent integrals of the n-body problem, which are algebraic with respect to q,p and t are the 10 described above.

(This theorem was later generalised by Poincaré). These results however do not imply that there does not exist a general solution of the n-body problem or that the perturbation series (Linstedt series) diverges. Indeed Sundman provided such a solution by means of convergent series. (See #Sundman's theorem for the 3-body problem).

is it proven that you cannot solve the n-body problem analytically?

Euler proved that the n body problem (n>2) is unsolvable analytically

We don't even know what the interactions look like exactly for more than 2 bodies. Take for example Helium

tim_lou said:
I heard that Sundman has a complete solution to the 3 problem using convergent series... even though it converges soooo slowly...

Can someone post the equation Sundman came up with? (even though i may not really understand it...) or informatios about the solution to the three body problem?

Sundmann showed that an integral power series representation must exist in terms of the inverses of the cube roots of the radial distances must exist. The French Academy of Science awarded Sundmann the de Pontécoulant's Prize for his work on solving the N-body problem. See this http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1915Obs...38..429.&data_type=PDF_HIGH&type=PRINTER&filetype=.pdf" .

The series converges very slowly due to the uncountable number of poles in the problem domain (any path with a collision sometime in the future creates a pole in the expansion). Since the masses are modeled as point masses, the paths involving collisions has measure zero.

loop quantum gravity said:
is it proven that you cannot solve the n-body problem analytically?
Be careful when you say things like this. When mathematicians say some problem is "insoluble", they mean insoluble in terms of some limited set of functions and some limited set of operations on those functions.

From Marion, J.B., "Classical Dynamics of Particles and Systems: Second Edition", Academic Press, New York, 1970
The addition of a third body to the system, however in general renders the problem insoluble in finite terms by means of any elementary function.

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Epicurus said:
We don't even know what the interactions look like exactly for more than 2 bodies. Take for example Helium

I'm not sure what you're talking about. I can certainly write down the three-body interaction for helium, it just involves three coulomb-type terms instead of the one for the two-body problem.

## 1. What is the Three Body Problem?

The Three Body Problem refers to a mathematical problem that involves the gravitational interactions between three bodies in space. It is a classic problem in physics and astronomy that has been studied for centuries.

## 2. Has the Three Body Problem ever been solved?

No, the Three Body Problem has not been solved in a general sense. This means that there is no one formula or solution that can accurately predict the motion of three bodies under the influence of gravity. However, there have been specific cases where the problem has been solved, such as when the bodies are evenly spaced and have equal masses.

## 3. Why is the Three Body Problem considered unsolvable?

The Three Body Problem is considered unsolvable because it involves three bodies with a complex interplay of gravitational forces. This leads to chaotic and unpredictable behavior, making it impossible to find a single formula or solution that can accurately describe the motion of the bodies over time.

## 4. What are some approaches to solving the Three Body Problem?

There are several approaches to solving the Three Body Problem, including numerical simulations, perturbation theory, and the use of computer algorithms. These methods can provide approximate solutions that can help us better understand the behavior of the three bodies in a given system.

## 5. How does solving the Three Body Problem impact our understanding of the universe?

Studying and attempting to solve the Three Body Problem has greatly contributed to our understanding of the laws of physics and the behavior of objects in space. It has also helped us develop new mathematical and computational methods, which have practical applications in fields such as astronomy, engineering, and computer science.

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