# The Three Body Problem

1. Oct 23, 2006

### 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?

2. Oct 24, 2006

### James R

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.

3. Oct 24, 2006

### StatMechGuy

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.

4. Oct 25, 2006

### tim_lou

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

Last edited: Oct 26, 2006
5. Oct 26, 2006

### student85

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

6. Oct 27, 2006

### StatMechGuy

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.

7. Oct 27, 2006

### pervect

Staff Emeritus
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:

8. Oct 28, 2006

### MathematicalPhysicist

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

9. Oct 28, 2006

### quinn

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

10. Oct 30, 2006

### Epicurus

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

11. Nov 13, 2006

### D H

Staff Emeritus
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" [Broken].

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.

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.

Last edited by a moderator: May 2, 2017
12. Nov 13, 2006

### StatMechGuy

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.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook