Solving the Impossible: Exploring 3-Body Mechanical Problems

[SpY]]

I've heard that it's apparently impossible to solve a 3-body mechanical problem, but I'd just like to know why. I'm told there are so many integrals and n'th order differential equations that you can't find an analytic solution, only a numerical one, but I want to know is it physically impossible, or just too difficult? I also know it's to do with chaos theory - that a small change in the initial condition makes a huge change in the whole system. But can supercomputers do it?

For instance to find the motion between the sun, Earth and moon simultaneously.

 

[D H]

The three body problem does not have a general solution in the elementary functions. This is a well-known result -- and so what? Lots and lots and lots of differential equations do not have solutions in the elementary functions. All that means is the things we have somewhat arbitrarily decided to denote as the elementary functions are not sufficiently powerful to describe a lot of differential equations.

As far as the three body problem is concerned, there is a series solution discovered about 100 years ago -- and it is pretty much worthless. There is nothing wrong with numerical solutions. Even the elementary functions require numerical solutions. What is sin(1)?

 

[lalbatros]

SpY],

What you read is probably a journalist's view on the KAM theorem and the "small divisors" problem in classical mechanics (like the htree body system).

You can easily find more material on the web about what the "small divisors" problem is.
Essentially it means that the pertubation theory fails to give convergent solution series near resonnances. This problem cannot be overcome by using "better methods", nor numerical methods. It actually implies, that in some circumstances you need a huge amount of information (numerical precision) to predict some outcome to questions like "is the solar system stable". Practically this sets a limit on predictability since we never know the inititial condition with an arbitrary precision. However, the importance of this problem may depend on the system under study: for example it is less acute for the sun-earth-moon system than for the whole solar system.

Please not that the precision of the calculation does not wipe out the resonance effect (or "the problem").
It still remains true that small changes in the initial conditions could change drastically the outcome.
It is furthermore alway possible to find a problem that will defeat any computer in this respect.
This does not imply that we cannot understand such sistuations by other means that simply calcualting the trajectories with ever increasing precision.

I can't tell you more about the KAM since this is known to be a very difficult topic and it far beyond my mathematical abilities.

Michel