# Three Body Problem

## Main Question or Discussion Point

I won't describe the problem, on the assumption that the people who know the answer to my question will already know what the problem is. It's sometimes called the three orbit problem, and it's a special case of the n-Body Problem.

There's a Wikipedia article about it, though it's not a very good article.

My question:

Is it more correct to say that the problem has been shown forever insoluble?

Or is it more correct to say that it cannot be solved by any known mathematical method?

(I understand there are special cases which are soluble. I understand that it can be "solved" by successive approximations, which grow increasingly inaccurate as the number of iterations increases.)

I'm not a mathematician or astronomer. I'm writing a book, in which I want to use this as an analogy for other things. I want to get it right.

It's my understanding that quantum computers, if they ever exist, will not have infinite computing power. If so, as far as I know, the three body problem will remain insoluble. Correct?

Isaac

Related Astronomy and Astrophysics News on Phys.org
Filip Larsen
Gold Member
Welcome to PF, mr. Newton.

Unlike the two-body problem, it is not possible to find an analytical solution to the three-body problem, that is a solution that would allow you to calculate the position of the bodies as a function of time for some given initial conditions. With various restrictions in the degrees of freedom in a three-body system you can derive some characterization of the possible orbits, but nothing that is considered an analytical solution.

However, it is not especially hard to analyze and solve the three-body (or indeed the n-body) problem if we turn to numerical methods. Such analysis can even include much more complex models (non-uniform planets, non-gravitational forces) that allow for all sorts of special perturbations to be included, so in terms of being able to accurately model and predict, say, the motion of the bodies in the solar system, the computers of today are quite up to the task. I'm not current on the latest research in the area, but to me it seems that the challenge today in getting more accurate celestial science is not the lack of computing power but the uncertainty regarding the precise details of relevant physical laws and measured data.

Last edited:
Welcome to PF, mr. Newton.
However, it is not especially hard to analyze and solve the three-body (or indeed the n-body) problem if we turn to numerical methods. Such analysis can even include much more complex models (non-uniform planets, non-gravitational forces) that allow for all sorts of special perturbations to be included, so in terms of being able to accurately model and predict, say, the motion of the bodies in the solar system, the computers of today are quite up to the task. I'm not current on the latest research in the area, but to me it seems that the challenge today in getting more accurate celestial science is not the lack of computing power but the uncertainty regarding the precise details of relevant physical laws and measured data.
Thank you, Filip.

If I'm not mistaken, you're talking about the method of successive iterations. My understanding is that this is reasonably accurate for millions of years into the future, but beyond a certain number of iterations, adding up to many millions of years, predictions about the future position and velocities of the planets in the solar system become hopelessly inaccurate.

Have I misunderstood?

Cheers,

Isaac

Filip Larsen
Gold Member
If I'm not mistaken, you're talking about the method of successive iterations. My understanding is that this is reasonably accurate for millions of years into the future, but beyond a certain number of iterations, adding up to many millions of years, predictions about the future position and velocities of the planets in the solar system become hopelessly inaccurate.

Have I misunderstood?
That is correct. Depending on what you want to analyse and how accurate your model and data is, there is a limit to how far into the future you can accurately predict trajectories. In general, both purely gravitational systems and systems which has friction (which in case of n-body simulation could be gravitational energy that ends up as heat in the core of moons and planets) has the possibility for so-called chaotic dynamics.

As you may know from simulation of the weather, presence of chaos effectively limits how far into the future you can predict individual trajectories since chaotic motion will amplify any microscopic inaccuracies in initial data to macroscopic differences later on. The same goes with gravitational systems although the time-frame here is much longer. The possibility of chaos in a system also mean that there is no real hope of finding a simple analytical solution for the motion since such solutions would be incapable of reproducing the structure found in chaotic motion.

Last edited:
That is correct. Depending on what you want to analyse and how accurate your model and data is, there is a limit to how far into the future you can accurately predict trajectories. In general, both purely gravitational systems and systems which has friction (which in case of n-body simulation could be gravitational energy that ends up as heat in the core of moons and planets) has the possibility for so-called chaotic dynamics.
Hello Filip,

I have always had the impression that the three-body problem is inherently chaotic, even in the absence of friction, even if the starting positions, masses and trajectories are known precisely. Is this incorrect?

Isaac

I have always had the impression that the three-body problem is inherently chaotic, even in the absence of friction, even if the starting positions, masses and trajectories are known precisely. Is this incorrect?
Depends on the exact type of problem. There are situations with the three body problem which are not chaotic, and there appear to be situations that are. If you have two objects in a circular orbit, and a small satellite that doesn't affect the two other bodies, you don't have chaos. My intuition is that in the general situation where you have two planets and a satellite, you don't get chaos.

Showing that a system is or is not chaotic is sometimes quite difficult, and there is a lot of interesting math involved.

Is it more correct to say that the problem has been shown forever insoluble?
It depends what you mean by "solve".

It's my understanding that quantum computers, if they ever exist, will not have infinite computing power. If so, as far as I know, the three body problem will remain insoluble. Correct?
I think that someone has come up with a solution to the three body problem that involves infinite series. The trouble is that once you've come up a three body solution, you add a fourth body, and that causes things to be really complicated.

Also there is a difference between "we don't know how to do this" and "we can prove that it can't be done." A lot of math involves proving that something just cannot be done. For example, Galois proved that you cannot solve a fifth degree polynomial function with a finite number of basic algebra steps.

In the situation with celestial mechanics, things are complex enough so that proving that you *can't* solve the problem for a given definition of solve gets you quite interesting mathematics.

One other idea that is important in chaotic systems is something called the Lynapanov exponent. Basically that number gives you an idea of how quickly the system blows up. I think the number for the solar system is five million years.

The other curious thing about the solar system is that the planets appear to be "chaotic but stable." What seems to be the situation is that the solar system is set up so that you can't figure out exactly where the planets are going to be over long periods of time, but it's set up in a way that you can be certain that planets are going to end up flying out of the solar system.

Filip Larsen
Gold Member
The other curious thing about the solar system is that the planets appear to be "chaotic but stable." What seems to be the situation is that the solar system is set up so that you can't figure out exactly where the planets are going to be over long periods of time, but it's set up in a way that you can be certain that planets are going to end up flying out of the solar system.
I don't think that the presence of chaotic orbits in the region of the planets current state (position and velocity) imply that a planet have to escape the solar system at some point in future time, i.e. that "chaos" implies "blow up", if that is what you mean. As far as I recall, it should be possible to have a solar system where planets in chaotic orbit are bounded in non-overlapping finites regions. Whether that is true or not for our solar system, I don't know. I guess the problem with predicting the precise state of our solar system on a very long term basis, is that it really is not a closed system, but may be affected from (yet unknown) objects from, say, the Oort cloud, nearby stars or similar.

alxm
Also there is a difference between "we don't know how to do this" and "we can prove that it can't be done." A lot of math involves proving that something just cannot be done. For example, Galois proved that you cannot solve a fifth degree polynomial function with a finite number of basic algebra steps.
And there is also analytical vs numerical. A fifth degree polynomial is unsolvable analytically in that way, but quite easy to solve numerically.

Likewise, the many-body problem may have an analytical solution that we don't know about, but it's entirely possible (perhaps even likely) that this analytical solution is more difficult to calculate than a numerical solution, in which case it's of fairly little practical use.

D H
Staff Emeritus
It depends what you mean by "solve".
Exactly. Saying that the three body problem is insoluble is rather sloppy. What people really mean when they say that is that the three body problem is insoluble in terms of elementary methods. What constitutes elementary methods is a bit arbitrary and is a bit meaningless. The trig functions are elementary, but that does not mean we can compute an exact values for, say, sin(1) or acos(0).

I think that someone has come up with a solution to the three body problem that involves infinite series.
That was Karl Sundman, and it's pretty much useless. The problem is that the series converges very, very slowly. The three body problem has singularities (there are lots of ways bodies can collide). While those singularities are a space of measure zero, they mess up the series a lot.

I wonder if this might be helpful for this discussion, on March 18, 2010 there was a full press release for the First Clay Mathematics Institute Millennium Prize For Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman.
Full press release: http://www.claymath.org/poincare/millenniumPrizeFull.pdf [Broken]
http://www.claymath.org/

Congratulations Dr. Perelman.

Last edited by a moderator:
D H
Staff Emeritus
The only connection between the Poincaré Conjecture and the n-body problem is that Henri Poincaré worked on both. He worked on a lot of very distinct problems.

D H, I was only addressing *your* mention of "singularities" as was in the pdf I earlier presented.

Perelman's breakthrough proof of the Poincaré conjecture was made possible by a number of new elements. He achieved a complete understanding of singularity formation in Ricci flow, as well as the way parts of the shape collapse onto lower-dimensional spaces. He introduced a new quantity, the entropy, which instead of measuring disorder at the atomic level, as in the classical theory of heat exchange, measures disorder in the global geometry of the space. This new entropy, like the thermodynamic quantity, increases as time passes. Perelman also introduced a related local quantity, the L-functional, and he used the theories originated by Cheeger and Aleksandrov to understand limits of spaces changing under Ricci flow. He showed that the time between formation of singularities could not become smaller and smaller, with singularities becoming spaced so closely – infinitesimally close – that the Ricci flow method would no longer apply. Perelman deployed his new ideas and methods with great technical mastery and described the results he obtained with elegant brevity. Mathematics has been deeply enriched.

D H
Staff Emeritus
FFS, ViewsofMars. Singularities can appear throughout mathematics. Just because they do appear in two areas does not mean that those two areas have anything to do with one another.

My reponse is on the next page.LOL! ( I wonder if one of those pranksters is up to mischief today!) (tee hee, I love it.)

Last edited:
FFS, ViewsofMars. Singularities can appear throughout mathematics. Just because they do appear in two areas does not mean that those two areas have anything to do with one another.
FFS, D H

"Singularities are to Ricci flow what black holes are to the evolution of the cosmos. Perelman also introduced a kind of geometric entropy, akin to the entropy studied in the exchange of heat, as in a turbine or motor."(Please refer to the link from Ereaka Alert that I earlier presented.) I provided an award given to Perelman who did discuss sigularities. Perhaps you would like to explain sigularities by way of a link (url) in particular the issue of two areas that don't have anything to do with the another.

Also, let's remember the OP Isaac Newton did earlier state, "I'm not a mathematician or astronomer. I'm writing a book, in which I want to use this as an analogy for other things. I want to get it right." His statement may imply an open door policy here on this topic. Math and astronomy, eh? He is writing a book and may wish to include both. Also, Isaac did say, "It's sometimes called the three orbit problem, and it's a special case of the n-Body Problem." The topic is "Three Body Problem."

I need to do my own research on "the three orbit problem", "n-body problem", and "three body problem." Foremost, do we have a problem?

I also have to remember that information changes quite fast in the realm of Science and Math. I do recall in Astromony or Cosmology there is mention of singularity or singularities. I have to hunt for that one.

People do love to learn. Life is a dance. It's fun!

Have a nice day,

Mars

Last edited:
@ViewsofMars: 1.) Why the double post?

@ViewsofMars & D H2.)... what is "FFS" ?

@ViewsofMars: 1.) Why the double post?

@ViewsofMars & D H2.)... what is "FFS" ?
I don't know!! I just now saw it. Weird. I was trying to make a few corrections on it and it appeared on this page. I'm going to try to erase the one on the previous page. Sorry.

Ok, I fixed it. I have to run or I'm going to miss my appointment.

Last edited:
@ViewsofMars & D H2.)... what is "FFS" ?
Frame Dragger, FFS means "FOR FURTHER STUDY."

D H
Staff Emeritus
ViewsofMars, 1/x has a singularity at x=0. That singularity has nothing to do with the Poincaré Conjecture.

FrameDragger, http://en.wikipedia.org/wiki/FFS. You decide.

As far as the three body problem is concerned, the oft-cited claim that it is insoluble is essentially wrong. The correct claim is that it is insoluble by means of elementary techniques. In that sense, so what? $\int_{-\infty}^x \exp(-t^2) dt$ is insoluble by elementary techniques, as are the solutions to host of differential equations (e.g., the gamma function, Bessel functions, ...). That these integrals cannot be expressed in terms of elementary functions does not mean we have to throw up our hands in despair. Numerical techniques can do quite a nice job solving these non-elementary integrals.

That said, the n-body problem appears to be chaotic. That means that numerical integration can only take us so far into the future. The numerical integration will yield a definite answer, but that result will essentially be garbage if the integration period is too long. The solar systems Lyapunov time appears to be 5-10 million years. Using numerical techniques on that kind of time scale will yield meaningless nonsense. The distances from the Sun might be close to correct, but where the planets lie on their orbits will be very wrong.

Frame Dragger, FFS means "FOR FURTHER STUDY."
Ahhh... ok... I'll um, take your word on that, even though another definition occured to me after posting.

A friend is working on the 2-body problem (Numerical GR), and his take on the 3+ is that it's utterly impossible at this time, or any concievable future time. *shrug* I'd say that's a pretty big problem! From what he's said, the 2-body is amazingly complex, and rooted in the need for perfect initial conditions and a LOOOOOT of HPDE's.

EDIT: @D H: OH. So that other definition is applicaple. I am backing away very slowly now!

D H
Staff Emeritus
The three body problem is not utterly impossible at this time. Think about it this way: How could we have sent vehicles to the Moon or other planets if it were utterly impossible?

The three body problem is not utterly impossible at this time. Think about it this way: How could we have sent vehicles to the Moon or other planets if it were utterly impossible?
I should clarify: impossible to find an exact solution over any given time-scale a la the 2-body problem. I should also add that something may be lost in the translation between my friend, and here, or he may have simplified matters for my sake.

D H
Staff Emeritus