Undergrad doing work on the 3 body problem.

[xdrgnh]

Let me cut to the chase. What are the chances a undergrad who took junior level mechanic class can make useful original research on the 3 body problem? If my understanding is right it hasn't been solved yet. I've been studying from Landua and studying from Taylor and just finished learning about the two body problem.

 

[twofish-quant]

Let me cut to the chase. What are the chances a undergrad who took junior level mechanic class can make useful original research on the 3 body problem? If my understanding is right it hasn't been solved yet. I've been studying from Landua and studying from Taylor and just finished learning about the two body problem.

If you have a good adviser, I think that you can find a doable topic.

Don't try solving the whole thing, but I'm pretty sure that there is some subset that you can do useful work on. A lot of dealing with the three model problem involves working on simplifications of it, and there are so many different variables that it's likely that there is something that no one has looked at.

The other thing that you can do is to try to apply some general theory to a specific situation. Or take known theory and then do something computation. One general algorithm for finding new projects is to add the words "with parallel computing/GPU". (I.e. we know how to make milkshakes, let's research how to make milkshakes with parallel computing/GPU")

 

[xdrgnh]

So we know how convert the two body problem into a one dimensional problem by using the CM frame. Now let's show that using parallel computing. Is that doing it right if I was doing the two body problem?. I'm seriously considering making that my goal over the rest of the summer and next semester doing work on the 3 body problem. I never had an adviser before so I assume the procedure to find an adviser would be approaching a professor I am close with regarding my desire to work on the 3 body problem.

Thanks a lot for your response.

 

[eri]

Have you done any research on the three body problem? There are solutions to it (in the form of power series) but they're generally useless for analytic work (they don't converge very quickly). That's why we make approximations. There are some quite sophisticated codes out there doing n-body simulations.

 

[D H]

If my understanding is right it hasn't been solved yet.

There are special case solutions of the restricted three body problem at the libration points and as pseudo orbits about the libration points.

Trying to "solve" the general three body problem is a fool's errand. It is provably "unsolvable". What that means is that the general three body problem cannot be solved as a closed form solution in terms of the elementary functions.

There is an infinite series solution, Sundman's series, but it's completely useless. On the order of 10^8,000,000 terms would be needed to attain a useful level of accuracy.

So that leaves numerical techniques for the general solution to the three body problem (and beyond). They work quite nicely, and you don't need 10^8,000,000 terms.

 

[xdrgnh]

Well my understanding is that the 3 body problem is a chaotic system which ties in with the nth body problem. I don't have any illusions of finding analytical solutions. Also just because they can't be solved using elementary function doesn't mean they can't be solved using special functions. Maybe some kind of intuition is missing.

 

[cgk]

So... what exactly is the advantage of solving it using special functions compared to numerically solving it using standard ODE integration techniques? [1] Also, technically, the above mentioned Sundman series /is/ a special function (I couldn't find information on it atm). Imo, even if you could find a reformulation, there is little benefit in replacing one horrible special function by another one...

I think it would be wise to choose a topic (i) which has not been under scunity for hundreds of years by hundreds of highly skilled mathematicians, (ii) which's solution would actually have impact if solved. Setting good goals is important.

[1] On a more philosophical level: Using ODE techniques, you can get a solution to any degree of accuracy you like (at least theoretically). And this is effectively the definition of "exact solution": You need to know a process which will give you a solution to a certain accuracy if requested. Even on a practical level this is not really different from writing down some special functions in terms of a series, which you also have to evaluate numerically (if you can do it at all).
This "exact solution" property actually even applies to the so-called "elementary functions" like sin, cos, etc.---there is nothing "elementary" about them. The process of evaluating such a function at a certain point is technically not different from evaluating anything else. E.g., erf (error function) is a non-elementary function. Evaluating it at a certain point or making a finite approximation to a certain degree of accuracy (as using, e.g., in math libraries) is no more simple or complex than for the exponential function or tangent function or similar.

 

[xdrgnh]

I just finished my freshmen year and the most math I took is complex variables. I choose the 3 body problem because the mathematics of it seems within my range and it coincides with the physics I've done. I know w.e discover if any I make in the 3 body problem won't be ground breaking but it might go in a journal and will be good for my career overall. But I got an idea of what I might want to try which I'll do next week once I finish up on doing some exam for my job.

Note

When it comes to the genius's who worked on the 3 body problem I will be standing on there shoulders hopefully that helps.

 

[AlephZero]

On a more philosophical level: Using ODE techniques, you can get a solution to any degree of accuracy you like (at least theoretically). And this is effectively the definition of "exact solution": You need to know a process which will give you a solution to a certain accuracy if requested.

Your "at least theoretically" papers over a very big crack in the logic here, unless you can PROVE that the numerical solution converges to the same solution as the ODE. That is not too easy if you don't have an solution to the ODE, and/or if its solutiosn can be chaotic in some situations.

I was once involved with similar question about modelling the lubrication of bearings. The existence of "oil whip" is well known (it results in a quasi-periodic motion at a frequency that is indepedent of any external forcing frequency, the shaft RPM, etc). Numerical simulations were giving results similar to oil whip but at different frequecies, with much debate as to whether these were phyiscally possible (which had some significant implcations if it was true) or just an artefact of the numerical methods. It took several years to resolve the question to everbody's satisfaction - and that was only a "2 body problem", though the ODE was a bit more complicated than Newton's laws of motion and gravitation.

 

[twofish-quant]

This conversation shows why it's important to get a good research adviser. You need someone that is working in the field that can tell you what problems are interesting. If you don't have someone close by that is actively doing research, then you'll need to learn how to use a research library to find and read journal articles, and then find someone that has some research background to keep you from going down a blind alley.

The reason I think this is doable (unlike say quantum field theory) is that the language of the three body problem is differential equations and power series, and so if you go into the literature you can read about what's been done. The main point of an adviser is that he or she can point you to which papers to read.

The other thing is that a lot of research projects involve *applying* a theory rather than developing one. For example, it may be that you have a researcher in exoplanets that wants to calculate the possible orbits of a planet around a binary system. In that situation you be trying to do your best to avoid doing anything creative mathematically with the n-body problem, since you are interested in the theory of exoplanets rather than the theory of n-body orbits.

In situations like that if you can download some computer code from the internet and use it, that's great, but you usually need non-trivial amounts of knowledge to make the code work.

One final thing. It's usually a bad idea to be too specific with what you want to research. A lot of what you end up researching depends on the expertise of the people around you, and it's a good idea to be flexible. If you happen to be near someone that is doing GR research, then you might end up doing the two GR body problem instead of the three body problem.

 

[cgk]

Your "at least theoretically" papers over a very big crack in the logic here, unless you can PROVE that the numerical solution converges to the same solution as the ODE. That is not too easy if you don't have an solution to the ODE, and/or if its solutiosn can be chaotic in some situations.

The reason I said "at least theoretically" is that in theory there is always a way of doing it numerically right. That's because the ODE if effectively *defined* in terms of a numerical process (or rather, it's limit in time step -> 0; and yes, applies only to ODEs of sufficient smoothness). So to get a more accurate solution, you just have to increase the number precision and decrease the time step.

Of course for a given requested accuracy, depending on the system (chaos, singularities etc) this can lead to an exponentially increasing computation time per simulation time, but mathematicians don't usually bother with that. I think there is little doubt that the process is "theoretically" guaranteed to give you an exact numerical solution, including proofs with various bounds (although I don't remember the names of any of the corresponding theorems).

I was once involved with similar question about modelling the lubrication of bearings. [...] Numerical simulations were giving results similar to oil whip but at different frequecies, with much debate as to whether these were phyiscally possible (which had some significant implcations if it was true) or just an artefact of the numerical methods.

That is a problem you might have with analytical solutions, too. Just that your solution is analytical does by no means mean that you can actually evaluate it. Just look into how much fun it is to calculate the various hypergeometric functions and related beasts... (btw: in the beginning, Mathematica had errors in them due to broken numerics...).

 

[D H]

Your "at least theoretically" papers over a very big crack in the logic here, unless you can PROVE that the numerical solution converges to the same solution as the ODE. That is not too easy if you don't have an solution to the ODE, and/or if its solutiosn can be chaotic in some situations.

cgk was correct. The only crack is Lipschitz continuity, which Newtonian gravitation obeys except at point singularities, as does linearized general relativity outside event horizons. A unique solution will exist so long as the trajectory avoids those singularities, and numerical techniques will (theoretically) converge to that solution.

I was once involved with similar question about modelling the lubrication of bearings. The existence of "oil whip" is well known (it results in a quasi-periodic motion at a frequency that is indepedent of any external forcing frequency, the shaft RPM, etc). Numerical simulations were giving results similar to oil whip but at different frequecies, with much debate as to whether these were phyiscally possible (which had some significant implcations if it was true) or just an artefact of the numerical methods. It took several years to resolve the question to everbody's satisfaction - and that was only a "2 body problem", though the ODE was a bit more complicated than Newton's laws of motion and gravitation.

What you are describing is neither an ODE nor a two body problem. As an N-body problem, this is an Avogadro's number quantum mechanics body problem. Nobody does that because this is unsolvable by any existing computational machinery. Instead one uses continuum mechanics to model the fluid, making the problem a boundary value PDE (rather than an ODE). This is pretty much insoluble as well, so one resorts to CFD techniques to approximate the solution to the PDE. Your problem was most likely one of the mathematical model not quite reflecting reality rather than the numerical techniques failing.

 

[D H]

Getting back on topic,

xdrgnh, twofish-quant's post on finding a research adviser was very good. A research advisor would tell you not to bother with trying to find a general solution to the three body problem. It's a fool's errand. What a research advisor would do would be to point you to papers/books on topics such as the Jacobi integral, invariant manifolds, halo orbits and Lissajous orbits, weak stability boundary trajectories, etc. These are the subjects of the interesting work in the solar system N-body problem. There's lots of interest in this problem because it saves a lot, a whole lot, of fuel. The Grail satellites currently orbiting the Moon got to that orbit using a weak stability boundary trajectory.

Or the advisor might ask you to look into examining how weak stability boundaries help explain the giant impactor model of the formation of the Moon. Or maybe he would ask you look outside the solar system at circumbinary planets such as Kepler-16b. Or she might ask to step a bit back from the three body problem and look into the numerical integration techniques that are so important to analyzing N-body problems. There's still lots of work being done on improved numerical integration techniques.

As twofish noted, it's best not to be too specific in your research interests. A good advisor will know what is of current research interest, and what a senior is capable of doing.

 

[AlephZero]

cgk was correct. The only crack is Lipschitz continuity, which Newtonian gravitation obeys except at point singularities, as does linearized general relativity outside event horizons. A unique solution will exist so long as the trajectory avoids those singularities, and numerical techniques will (theoretically) converge to that solution.

I don't have a problem with that as a piece of pure math, but whether it applies to numerical conputations using approximate arithimetic is a different question IMO. But I don't want to hijack this thread!

Getting back on topic,
Or she might ask to step a bit back from the three body problem and look into the numerical integration techniques that are so important to analyzing N-body problems. There's still lots of work being done on improved numerical integration techniques.

I definitely concur with that, whatever our philosophical differences on the relevance of QM to Mech Eng

 

[twofish-quant]

What a research advisor would do would be to point you to papers/books on topics such as the Jacobi integral, invariant manifolds, halo orbits and Lissajous orbits, weak stability boundary trajectories, etc.

More to the point, a research adviser will likely point you to a small subset of these topics. One of the basic mistakes that undergraduates make is to try to explain the universe. The trouble is that the universe is too big and complex.

However, big and complex means lots of research opportunities. The good news is that the three body problem is so big and complex that it's very likely that there is some particular aspect of it that no one has looked at. That's were you come in. A lot of the problems that research advisers have undergraduates look at are of the form "I wish I had time to calculate (fill in the blank)."

One other thing, looking for a research adviser is something that you'll need to do even much later in your career. I don't have any particular expertise in orbital mechanics, and I'm essentially telling you what I'd do if I wanted to do work in the three-body problem.

Something else that is sort of cool is that it doesn't take much better to become an expert in a small problem. Once you get pointed in the right direction, with about two to three weeks of work, you'll know more about your subtopic than most of the people on the list. With about two or three months of work, you'll likely know more about the subtopic than your adviser.