Could Virtual Particles Solve the N-Body Problem?

[CosmicKitten]

N-Body Problem -- ideas?

First of all I would like some explanation on why one can't find a GENERAL solution for the N-body problem, and what exactly we have to use in its place.

Secondly, I want to know if anybody understands these vague ideas of mine and how they would work...

Let's say, for each particle in the system, we could imagine a "virtual particle" (which has nothing to do with the virtual particles from other branches of physics, which I know nothing about) whose motion could be traced as a function of the initial positions, velocities, etc. of all of the other particles, and instead of regarding the given particle's motion as a function of all the other particles we could simply follow its motion with respect to the virtual particle, so it would be reduced to a 2-body problem or perhaps not even that, if we assume that the virtual particle's motion is unaffected by the actual particle that is following it. And perhaps we might need to make new rules on virtual particle motion, how they react with each other, do they fall in on each other, etc...

A virtual particle would be somewhat similar to center of mass, I think? Regarding the center of mass itself as a virtual particle, its velocity at anyone moment can be regarded as the vector sum of the velocities of all the other particles at that moment (not sure how relativity would play into it).

Sorry if this makes no sense...

 

[mfb]

The system is not integrable (hmm, the wikipedia page looks a bit complicated, I am sure there are better explanations somewhere).

How do you calculate the motion of your virtual particles? If you take the other virtual particles into account, you have the same problem again.
(And no, it will not work, this is a proven mathematical statement not our limited knowledge)

and what exactly we have to use in its place.

Numerical methods (especially: calculate momentum+position for each time step) or other approximations.

 

[CosmicKitten]

I already figured out a numerical method, that involved for each time step regarding the other bodies as stiff and not interacting with the one you are calculating the movement of, then for the next time step doing it over with respect to the new positions. And then taking the limit of the sum as the number of time steps goes to infinity and the length of the time step goes to zero... then I found out this method already existed and is called the Euler method, I believe? Mind you I was only in calc 2 when I figured it out, though I was convinced that it was below my level so I checked out harder books and got some ideas after a brief readthrough of Differential Equations Demystified... What exactly is the proven mathematical statement? And how is it proven?
Now let's start with a system where all the masses are the same. Say you drew the vector lines for force through each mass. Will these vector lines all intersect at the same point? I'm pretty sure it doesn't do so when the masses are different... anyway if they do and if this is true for all times then one could write an equation for the motion of each particle as following the center of mass like a tractrix without regard to the influences of the other particles...
If not, well pretty much any point that lies on the vector line of force for each particle could be taken to be its "local center of mass" (the point that it's following around, like a tractrix). So the vector line would be all of the possible solutions but one would have to find the one for each that works for the entire system. If there is no such solution then the bodies cannot move in such a way.
Also, suppose that one takes a point where any two or more force vectors intersect as a "local center of mass". Could one possibly find a way to track the motion of these? Would these points perhaps try to close in on each other so that there is only one center of mass? Viewed a different way, perhaps the system's ultimate goal is to establish an equilibrium where all of the local centers of mass intersect to one center of mass, and then further establish equilibrium by trying to get all the vectors on the same plane and all the vector "sources" (the masses) at the same point? Mind you they will TRY but in many cases never, as time goes to infinity, reach such a condition, just as an object in orbit tries to fall down but can't because it has too much energy...

 

[mfb]

The easiest numerical method: Start with time t=0 and some positions and velocities. Based on the current positions, calculate the acceleration of all objects, based on velocity and acceleration, calculate the velocity after a small timestep δt. Calculate the new position of all objects, based on the old position and old or new velocity or the average of both or something similar. Calculate the accelerations again and so on.
Those methods tend to accumulate errors after a while, and there are better methods. However, all methods will give some errors, the question is just how quick and how large.

What exactly is the proven mathematical statement? And how is it proven?

The system is not integrable and chaotic, see the wikipedia page or some books for details.

Now let's start with a system where all the masses are the same. Say you drew the vector lines for force through each mass. Will these vector lines all intersect at the same point?

No.

If not, well pretty much any point that lies on the vector line of force for each particle could be taken to be its "local center of mass" (the point that it's following around, like a tractrix). So the vector line would be all of the possible solutions but one would have to find the one for each that works for the entire system. If there is no such solution then the bodies cannot move in such a way.

??
The force vector is not a line. If you consider the line through an object, parallel to the force: In general, this is not the path of the object. And the line will shift and change its direction with time.

Also, suppose that one takes a point where any two or more force vectors intersect as a "local center of mass".

In general, you do not get any intersection of those lines. So everything based on those points does not help.

Viewed a different way, perhaps the system's ultimate goal is to establish an equilibrium where all of the local centers of mass intersect to one center of mass

No. An n-body system will not establish any equilibrium in the way you try to find it.
And by the way, those systems are not intelligent. They do not have goals or try to do something.

 

[CosmicKitten]

The goal is to solve the equations for the motion of these "virtual particles", which would thus solve the equations for the motion of the real particles, whose motion can be regarded as simply following these points around. The challenge would be to find an equation for such a setup that would result in the same motion that the entire system of particles would cause. I figure that such a method might help to solve some unsolvable cases, if not all of them? But I figure that to solve for the virtual particles would be just as hard as to solve for the actual particles, unless there is some mathematical trick I don't know about?

See to trace the motion of a real particle tagging a virtual particle would not even be a 2-body problem, for though the real particle is following the virtual particle according to gravitational laws, the virtual particle is not following it back. Unless one were to solve for the equation of a virtual particle that DOES follow it back, but I'm not sure which of the two would be harder or even possible.

How would one attempt to set up such an equation? I imagine equating the equation of the second derivative of the particle's position with respect to time to the position of the virtual particle multiplied by some constant... I might be terribly wrong here... basically set up so that the second derivative is the force vector and is always pointing toward this virtual particle no matter where it goes...

 

[SHISHKABOB]

I was under the impression that solving the N-body problem was technically impossible, since there are too many free parameters.

 

[D H]

I was under the impression that solving the N-body problem was technically impossible, since there are too many free parameters.

When physicists and mathematicians say the N-body problem isn't integrable, what they mean is that it isn't integrable in closed form in the elementary functions -- just as $\frac 2{\surd \pi}\exp(\frac{-x^2}2)$ isn't integrable in the elementary functions. Regarding the N-body problem, There are 6N degrees of freedom, three each for position and velocity for each of the N bodies. There are 6N easily calculable derivatives at any point in time. Given a set of positions and velocities at some point in time t₀, this set of differential equations is integrable in the sense that a unique solution exists for all t>t₀ -- except for a space of measure zero in that ℝ^6N phase space. In other words, except for those singular cases, an integral does exist; it just can't be expressed in closed form in terms of the elementary functions.

If a closed form solution can't be found, what about an infinite series? That approach certainly works with regard to integrating $\frac 2{\surd \pi}\exp(\frac{-x^2}2)$. This latter integral comes up so often that it's given a special name, the error function. This approach also works for the N-body problem. About 100 years ago, Karl Sundman found a series solution in t^-1/3 for the three body problem, and about twenty years ago Qiu-Dong Wang generalized this and found "The global solution of the n-body problem" (http://adsabs.harvard.edu/full/1991CeMDA..50...73W).

These series solutions unambiguously demonstrate that the N-body problem is soluble. However, there's a minor problem with these series solutions: They're utterly worthless numerically. Sundman's series, for example, requires 10^8,000,000 terms to get anywhere close to astronomical precision.

 

[mfb]

@CosmicKitten: Do you realize that your approach (or at least what you describe) does not help in any way? As far as I can see, you propose to replace the original problem by another problem, without any specification how the other problem differs from the original one. And then you think that the second problem is easier for some unknown reason.

 

[SHISHKABOB]

I see! Thank you very much. I guess that the closed form solution would be solving it like you would try to solve, say, a two-body problem. Like a set of simultaneous equations?

 

[CosmicKitten]

Now most integrals can also be expressed as a Riemann sum, you know, take the limit of the sum of all of the values for the time steps as the number of time steps that the function is split into approaches infinity. A similar sort of sum limit would be used for such a problem to approximate the motion right? Why could one not take the limit and find what the integral of the function could be?

For that matter why can't a new kind of 'elementary' function be created to express such problems? After all before e was discovered nobody knew of a function that is equal to its derivative/integral. Before imaginary numbers and trigonometric functions many other problems were unsolvable. If nobody had ever discovered logarithms who would ever have guessed that the integral of one over x could have a solution?

Are all of the derivatives of every order with respect to time continuous? I would assume so seeing that all of the forces are present from the start, though in the case that bodies collide the motion would come to an abrupt stop and a noncontinous graph but such a function would still be continuous if it would be assumed the bodies would just pass through one another?

Does anything pertaining to the "solvability" of the problem change when relativity is taken into account?

 

[D H]

For that matter why can't a new kind of 'elementary' function be created to express such problems?

First off, the concept of elementary functions is well-defined. There aren't any more than what we already have, by definition. What you are talking about would fall into the class of special functions, things such as the error function, to which I alluded in my previous post.

There's a much bigger problem, however. You can't find this special function. The series that describes it is, for all practical purposes, impossible to evaluate -- even for the three body problem. The best that can be done is to numerically integrate the equations of motion given an initial set of conditions.

 

[mfb]

Does anything pertaining to the "solvability" of the problem change when relativity is taken into account?

It does - in general relativity, even the general 2-body system has no simple solution.

 

[CosmicKitten]

Sorry to revive an OLD old thread, but I have been wondering lately about how a computer simulation to calculate (using approximations of course) which direction with a given speed an object at point A would have to take to get to point B, given the velocities, masses and positions of a number of other objects. Would it be more efficient to simultaneously calculate backwards from point B and at some time or some point in the middle, figure out the parameters for the other objects that would get the backwards from point B position to be equal to the forwards from point A position and from that calculate the velocity? Would there be a way to tell if there is no such velocity with the given magnitude and arrangement of other objects that will get from point A to point B within a certain time period? I'm guessing it's not possible to tell if it would get there given infinite time, since it's too chaotic to find the limit as the time goes to infinity?

Perhaps it would simplify calculations if the other objects could be lumped together as one object, with the velocity being equal to the vector sum of the velocities of all the original particles, and the mass being the reduced mass, or would that just make it more complicated?

 

[mfb]

Calculate a lot of trajectories, I think there is no easier way. A good program will have some clever way to decide which trajectories are interesting, but as long as you have more than one object with a relevant influence, I don't think you can find solutions without simulations. Forwards/backwards does not matter.

Given infinite time or unlimited velocity, I would guess the general case always has a solution unless obvious things make it impossible - like starting inside a black hole :p.

Perhaps it would simplify calculations if the other objects could be lumped together as one object, with the velocity being equal to the vector sum of the velocities of all the original particles, and the mass being the reduced mass, or would that just make it more complicated?

That does not work.

 

[AlephZero]

Another idea is to start with an arbitrary trajectory (e.g. a straight line from A to B at constant velocity) which is not consistent with the physics, because you have to apply some external force (which you can calculate as a function of time and/or position) to make the object follow that trajectory.

Then, find how the trajectory is perturbed as you gradually reduce the amount of external force to zero.

For an N-body model of a solar system, you could usually make a much better choice of initial trajectory than a straight line, of course.

 

[tfr000]

Perhaps it would simplify calculations if the other objects could be lumped together as one object, with the velocity being equal to the vector sum of the velocities of all the original particles, and the mass being the reduced mass, or would that just make it more complicated?

Well, lumping together is more or less what you're doing when you add all of the forces from all of the other objects. One could equate all of that with one force from one body. You might want to google "Cowell's method", which is the standard way that astronomers model the Solar System.

 

[ASG141]

what is a N body problem? i don't understand it.

 

[SteamKing]

what is a N body problem? i don't understand it.

It's one of the most famous unsolved problems in classical mechanics:

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

 

[Khashishi]

There's no issue with simulating an N-body problem numerically (beyond the usual issues of stability and precision). However, finding an analytic solution is going to be tough in general. You can use the center of mass to reduce the number of independent bodies by one (changing a two body problem into a one body problem), but you can't reduce it anymore, unless you have additional constraints. (The physics of the center of mass is a constraint.)

As far as reversing the motion, whether that strategy will work depends on the specifics of the problem. Basically, it depends on how quickly a deviation in initial conditions will grow as an error in the outcome, and the inverse question of how quickly a deviation in the outcome will grow into an error in initial conditions. It has to do with the Lyapunov exponent.