n-body problem

Hey everyone,

I thought I'd coin up this fun problem. I couldn't find the root of the problem in wikipedia. The problem states: "The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally.".

Now I just made a gravitational simulation in 3dgs with 200 masses roaming happily around... why, according to the problem, wouldn't I be able to determine their paths? If I couldn't predict (or calculate) the next position of each body, I would be unable to create the simulation in the first place...

You can't solve the problem analytically for n>=3. That is to say, you cannot solve the whole thing mathematically so that you get an exact function

r(t)

(place as a function of time, say)

for all n bodies.
Of course, nothing stops us to approximate (even to excellent precision) their paths, as for instance you did in your simulation (though I'd guess we can reach higher precision than your simulation did :)). But it is not perfect and exact, and over large timescales, you will see that your simulated prediction will differ to what you'd see in nature.


Does that answer your question?

as far as i understood the article, n-body problem cannot be solved analytically for n > 3, and you solve it numerically (not sure how to say it in English)

ah, error ninja'd me

So, what you say is that the precision of the next position is tied to the interval length between two timestamps? The smaller your interval, the more precise you approach reality. Digitally you can calculate this position to an exact precision, from T=0 to T=1, but with an analogue timelapse you can go to infinitely small intervals, T=0 to T=0.000..01, and since infinitely small is not reachable, you never know precisely where you end up. Makes sense.

Basically, yes: We're kinda "approximating" an integral by summing, but no matter how small we do our summation intervall, unless its infinitesimal (which is what we call "infinitely small" :)) we'll always do a mistake, and those tend to add up.



Other problems complicate things further: Such as the fact that the differential equations governing their motion are connected (i.e. you need to know the one of A to solve for B and vice-versa), and unlike in a two-body problem, it's not easy to "decouple" them (so that you only get differential equations for stuff that do not depend on each other anymore).

It's actually quite enlightening and a cool bit of cleverness to see how you do that for the two-body problem (Kepler). If you're not afraid of a bit of math, check it out!

This is actually the explanation of the theory of chaos
The theory of chaos and the 3 bodies problem are strictly related items but they are not exactly the same stuff
Put it this way
You have 3 bodies , the only forces involved are the gravitational forces
It is quite easy to write the differential equation, giving the position of the 3 bodies at a certain time T
Despite a common misconception these are deterministic problems
Given the initial configuration, there is one and one only possible configuration at time T
The point is that in most cases these equations can not be solved
As you know there is not a general method to solve differential equations

You can however try to guess the type of solution

Are the solutions closed ( such as a circle) or open ( such as a parabole) functions ?
In the former case the system is stable
After a initial transiction period the 3 bodies sooner or later will alwayes pass through the same points in the space
In the latter case it is not

Henry Poincare , at the beginning of tlast century found the solution using topology methods a branch of math which was his own creature
The solution of the differential equation can be open functions thus the system may not be stable

The link to the theory of chaose and the quoted claim is the following
Even very small differences in the initial parameters can make a "closed" solution turn into an "open" one

This was shock
Scientists gave for granted that a deterministic system is also predictable ( see the famous Laplace claim...)
From the 3 bodies problem the theory of chaos got started

that's damn interesting. maybe you can advice a couple of books/papers that give a basic idea of this topic? not the ones that google comes up with, but the ones that you find valuable

I am happy to see that these topics are of some interst

For the theory of chaos I would suggest the book :

"The computational beauty of nature " by Gary William Flake

It should be of particular interest for programmers

For the "n bodies problem" I can not be of help
I own the 25 books series "Mondo matematico " ( Mathematical world ) but it must be an Italian edition only
Please note however that this topic needs the knowledge of extremely advanced maths
I dont have myself the slightest idea how Poincarè achieved the results which I explained in my previous post

I would like just to add something more about this topic

Common sense assumes that a deterministic process must be , in principle, also predictable
It is not
A reason is the one explained by Error 014
An other reason is the n body problem itself

Why could Newton discover the gravitational laws ?
Because the 2 bodies Sun -Earth system entails a periodical orbit
Suppose that our earth belongs to a n bodies system
Actually it does but the influence of the other planets is negligible

The path of our earth in a time interval T , would be different than the one in the previous and in the next time interval T , regardless of T duration
In such a case It would be impossible to find out the laws of motion even though they are exactly the same as in a 2 body system

This was the shock caused by Poincare's analysis of the n body problem

I suppose this answer Joozey's questio

He said, I made a computer simulation
I can predict the configuration of 200 bodies why does wikipedia claim that it is not possible ?

Aside from the other consideration,which are correct ,the answer is
Because you know the law of motion that you have set
An observer watching your simulation could not predict the future even using the most sophisticated mathematical tools

Sounds vaguely similar to the "Principia Mathematica" series to me? Who are the authors?

I suppose it is an Italian edition only, various authors

http://www.rbaitalia.it/mondomatematico/collezione/

Looks very interesting and looks like a great introduction to the various mathematical topics, like prime numbers and such indeed. Does it go in depth with those topics or are topics discussed at a non-academic level?

Contrary to what I previously assumed, it doesn't seem like the Principia Mathematica, which isn't necessarily a bad thing if you actually want to understand more easily what it's about haha.

The right compromise in my opinion
Many scientific books targeting a large audience , quite often, make use of silly examples
Some thing like
" I am a great scientist( the author) you ( the reader) are
an illeterate so I teach you quantum physics same as a grandfather tell a tale to his five years nephew "

These books may skip the more complicated demonstration ,focusing on the results,but what they say is absolutely exact from a scientific point of view