Is the Three-Body Problem Unsolvable?

[Saketh]

I was reading the Encyclopedia Britannica today online, and found this statement under the Three-Body Problem article:

in astronomy, the problem of determining the motion of three celestial bodies moving under no influence other than that of their mutual gravitation. No general solution of this problem (or the more general problem involving more than three bodies) is possible.

I was curious, because I have seen other sources call it an "unsolved problem." Is this problem impossible to solve, or is it just that no one has solved it yet?

And if it is impossible, where can I find a proof of the impossibility?

Thanks.

 

[D H]

One can similarly say that \int e^{-x^2} dx is insoluble. When mathematicians say something is "insoluble" they almost always mean insoluble with respect to a limited set of functions.

Here is a better synopsis of the three body problem:
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.​

In other words, the solution is not an elementary function. The problem is soluble, just not using elementary functions. Sundmann showed that an power series representation of the N-body problem must exist, where the power series is in terms of the inverses of the cube roots of the radial distances. Sundmann was awarded the de Pontécoulant's Prize by the French Academy of Science for his work in solving the N-body problem. A 1915 review of his work is here: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1915Obs...38..429.&data_type=PDF_HIGH&type=PRINTER&filetype=.pdf"

Sundmann only proved existence. 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).Nobody uses Sundmann's approach to solve the N-body problem. They integrate the equations of motion.

 

[gone gator]

Here is a better synopsis of the three body problem:
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.​

In other words, the solution is not an elementary function. The problem is soluble, just not using elementary functions. Sundmann showed that an power series representation of the N-body problem must exist,
...
Nobody uses Sundmann's approach to solve the N-body problem. They integrate the equations of motion.

can you explain what you mean by "integrate" in that last sentence?

do you really believe anyone "solves" this problem, in a mathematician's sense of soluble? even Newton appealed to God here, for good reason. for almost any initial condition we know a unique solution exists, nothing more. no?

 

[D H]

can you explain what you mean by "integrate" in that last sentence?

I meant numerical integration, or more specifically, numerical solution of a multidimensional ordinary differential equation. A wiki article on this topic is http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations" .

do you really believe anyone "solves" this problem, in a mathematician's sense of soluble?

What do you mean by "solve"? What are all of the digits of \pi? No one, including mathematicians, has answered that particular question.

Sundmann showed a solution exists, which is http://www.xs4all.nl/~jcdverha/scijokes/6_2.html" .

 

[gone gator]

I meant numerical integration, or more specifically, numerical solution of a multidimensional ordinary differential equation.

not such a straightforward thing for chaotic ODEs (all of must be, of course, "multi-dimensional").

What do you mean by "solve"? What are all of the digits of \pi?

smells a bit like a red-herring here. i do not see how the specification of the digits of pi is algorithmically similar to numerical integration of a chaotic ODE on a digital computer. please help me if I've missed something obvious here!

Sundmann showed a solution exists good enough for most mathematicians.

in fact, nonconstructive proofs did not used to be good enough for mathematicians, and there are still many today who'd be happy with a bit more. or atleast to say something more than "a solution exists".

One can get as close to the "truth" as one wants with numerical integration. For an engineer or a physicist that is good enough.

i'd like to understand why you think this is true. (no trick question here, just true in the real world, something i could put in my thesis and defend.) what do you feel would be the most difficult part of actually "solving" the future position of the earth-moon, sun, Jupiter problem?

Newton, for instance, wanted to know if such a system was stable (i am happy to stay with the more simple system, leaving out the other planets): can you answer that question by "numerical integration"?

Does it make it easier if i relax the question of infinite time stability to one on the timescales before the sun expands and vapourises the earth-moon?

 

[D H]

To be brunt, that is a silly question. You are mixing mathematics and the real world. If the sun and planets were point masses, if no other objects existed in the universe, if we knew the exist states of those objects at some point in time, if Newton's Law of Gravity completely described the equations of motion, then the answer is yes. Mathematically.

The only way to truly assess "infinite time stability" of the solar system is to know the initial states of all the objects in the universe to infinite precision. The only way to assess the stability of the solar system until the "sun expands and vaporizes the earth" is an equally silly question. We don't know the current states of the planets to anywhere near the accuracy needed to do that.

We do know that, insofar as all of the objects in the solar system are concerned, the solar system is not stable. The collision of Comet Shoemaker-Levy and Jupiter is proof of that instability. On the other hand, the sun and planets (Pluto doesn't count) have been quasi-stable for billions of years. As far as engineers and physicists are concerned, this is "close enough" to proof that the sun and planets are in a stable configuration.

 

[Chris Hillman]

Just wanted to add very quickly that mathematicians have found some exact n-body problems including things like fascinating knotted orbits. These are of course very special solutions! Still, they are quite fascinating.

There also continues to be much work on understanding qualitative features of multibody dynamics in Newtonian gravitation. Some recent work suggests that we can be a lot more clever in planning interplantetary space exploration missions, for example. (Unmanned of course, since anyone rational enough to use mathematics will oppose manned spaceflight in favor of robotic missions.)

 

[Integral]

http://www.colorado.edu/physics/2000/applets/satellites.html" for some hands on 3 body fun.

 

[D H]

Just wanted to add very quickly that mathematicians have found some exact n-body problems including things like fascinating knotted orbits. These are of course very special solutions! Still, they are quite fascinating.

There also continues to be much work on understanding qualitative features of multibody dynamics in Newtonian gravitation. Some recent work suggests that we can be a lot more clever in planning interplantetary space exploration missions, for example. (Unmanned of course, since anyone rational enough to use mathematics will oppose manned spaceflight in favor of robotic missions.)

The first remark is right on the mark. Mathematicians have been studying the n-body problem for a long time. Lagrange found five special cases of the 3 body problem that have exact solutions.

The low energy transfers take advantage of the co-linear Lagrange points. JPL has already used these in planning and executing the SOHO, SMART-1, and Genesis missions.

The final off-topic remark is a pile of rubbish and is quite insulting to the thousands of people who are quite rational enough to use mathematics in their work every day on human spaceflight.

 

[gone gator]

To be brunt, that is a silly question.

perhaps, but then it has drawn the interest of many physicists and astronomers for several hundred years.

You are mixing mathematics and the real world.

guilty as charged: i think they call doing that "physics"

If the sun and planets were point masses, if no other objects existed in the universe, if we knew the exist states of those objects at some point in time, if Newton's Law of Gravity completely described the equations of motion, then the answer is yes.

since you did not specify the question you were responding to bruntly, i am not sure what question you claim has the answer "yes". but i expect (a) your list is incomplete and (b) the answer is "probably". what was the question?

i suggested a simpler question (point masses, fewer bodies, &c) as a common means of approaching a physics problem: first just see how hard it is to do an "easier" version of the problem. if the easier version in not tractable we learn something, and perhaps save some head banging (sometimes).

The only way to truly assess "infinite time stability" of the solar system is to know the initial states of all the objects in the universe to infinite precision.

this statement is, in general, false, although it might apply in special cases. but I moved to the "finite time" version to avoid confusion.

The only way to assess the stability of the solar system until the "sun expands and vaporizes the earth" is an equally silly question. We don't know the current states of the planets to anywhere near the accuracy needed to do that..

there are, of course, other ways to establish the stability of a system than forecasting its exact future. no? (for example if within a volume of state space, all initial conditions can be shown to be stable...)

We do know that, insofar as all of the objects in the solar system are concerned, the solar system is not stable. The collision of Comet Shoemaker-Levy and Jupiter is proof of that instability..

in that case, every meteor streak in the night sky is evidence, no?

On the other hand, the sun and planets (Pluto doesn't count) have been quasi-stable for billions of years. As far as engineers and physicists are concerned, this is "close enough" to proof that the sun and planets are in a stable configuration.

i cannot speak for enginneers, but i doubt that definition of close enough would satisfy many physics examiners or any reviewers for PRL. i'd submit the continued interest in the few-body problem over hundreds of years as evidence against your claim.

hope that was more clear, sorry it was so long. happy to try and clarify further.

 

[gone gator]

Just wanted to add very quickly that mathematicians have found some exact n-body problems including things like fascinating knotted orbits..

nitpick: since the dimension of the state space is somewhat greater than three, they are not, in fact, "knotted"

These are of course very special solutions! Still, they are quite fascinating.

agreed. (on both counts).