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From: ethan@utastro.UUCP (Ethan Vishniac)
Newsgroups: net.physics
Subject: Re: the multi-body problem
Message-ID: <757@utastro.UUCP>
Date: Thu, 26-Sep-85 10:25:46 EDT
Article-I.D.: utastro.757
Posted: Thu Sep 26 10:25:46 1985
Date-Received: Sun, 29-Sep-85 09:06:03 EDT
References: <1330@teddy.UUCP>
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Organization: U. Texas, Astronomy, Austin, TX
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Xref: tektronix net.physics:03454 

> []
> One often hears that the two-body problem (two bodies interacting
> gravitationally) is completely solvable (I guess that means that
> one can completely describe the motions and ineteractions of these
> two bodies in an isolated system), but when the problem involves any
> more tha two bodies (3 or "many"), then there does not exist a known
> solution for describing the system completely. About this I have several
> questions:
> 
>     1.	Why is the three- or many-bodied problem unsolvable? (Note I 
> 	realize that, given the asccuracy with which we can navigate
> 	about the solar system, then the problem, while unsolved, is
> 	approachable with some spectacularily good approximations).
> 
It is not unsolvable in the sense that when one is given initial conditions
it is not particularly difficult to calculate the future evolution of the
system indefinitely far into the future.  It is unsolvable in the formal sense
that the number of constants of motion is less than the number of constants
of integration in the problem.  Therefore the general features of the
evolution of the system can (apparrently) only be solved for by brute force.
The constants of motion are: angular momentum (three components), momentum
of the center of mass (three components), total energy of the system,
and total mass (however since the mass of each object is separately
conserved this last is trivial).

>     2.	Do the three-body problems apply for systems where the mass of
> 	one of the bodies is vanishingly small compared to the others
> 	(such as in a Voyager/Jupiter/Sun system)?
> 
In such a case it can be solved as a perturbation problem.  Difficulties
arise only after absurdly long times.

>     3.	Since general relativity seems to approach gravitation not as
> 	a force acting over a distance, but more as a deformation in the
> 	geometry of space-time (a wild simplification, I agree), can the
> 	three- (or many-) bodied problem be solved as a geometry problem?
> 	In other words, is the difficulty associated with a Newtonian
> 	view of gravity and the attendant mechanisms, or does general
> 	relativity suffer the same way?
> 
GR has the same problems here.

>     4.	Is the solution to all this merely one of computational 
> 	fortitude? (Has JPL solved the problem simply by brute
> 	force, or has the brute force merely made their approximations
> 	less approximate?)

See above.

> Dick Pierce

People have had *some* success finding approximate constants of motion
that help give some insight in the way the system moves around phase
space.
-- 
"Support the revolution        Ethan Vishniac
 in Latin America...           {charm,ut-sally,ut-ngp,noao}!utastro!ethan
    Buy Cocaine"               ethan@astro.UTEXAS.EDU
                               Department of Astronomy
                               University of Texas


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