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From: john@frog.UUCP (John Woods)
Newsgroups: net.physics
Subject: Re: the multi-body problem
Message-ID: <253@frog.UUCP>
Date: Tue, 1-Oct-85 17:59:54 EDT
Article-I.D.: frog.253
Posted: Tue Oct  1 17:59:54 1985
Date-Received: Fri, 4-Oct-85 01:38:14 EDT
References: <1330@teddy.UUCP>
Distribution: na
Organization: Charles River Data Systems, Framingham MA
Lines: 63

> 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:
>
What "they" really mean to say is that the two body problem is completely
*solved*, in that someone has taken the differential equations (which are
quite easy to write down for any arbitrary number of bodies) and integrated
them to produce equations of position as a function of time.  One can easily
write the differential equations for the 3-body problem, but of the many
titanic intellects who have spent time on the problem (Gauss, for instance),
not one has been able to arrive at the integrated equations for Xn(t).  This
is usually used, in Differential Equations courses, as the motivation for
numerical approximation.  So, 

>     1.	Why is the three- or many-bodied problem unsolvable?
It is not *known* to be unsolvable (yet).  It is known to be extremely tough.
If I remember correctly, there is a proven theorem in DE that *all*
differential equations have existant, unique solutions (unique after boundary
conditions are applied) -- which makes the lack of a solution to the three-
body problem even more irritating.  :-)

>     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)?
>
Yes, but it turns out that the special case of one mass being "zero" is as
easy to solve as the two body problem (in fact, it looks much like the two
body problem).

>     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?
General relativity, if anything, has more complicated equations.  As a wild
guess, solving the geometrical equations would be even worse a nightmare than
solving the Newtonian equations (otherwise, one would think that someone would
have already solved them).

>     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?)
Even the brute force available in a TI-59 calculator is sufficient to get
a "good" approximation (a "feel" for the orbits) in the 3 body problem.  Real
live brute force will get you (or your spacecraft) a *loooooong* way...

For those who have had calculus, I recommend any standard text on Differential
Equations for further study (sorry, I don't remember the name of the text I
used at MIT).

--
John Woods, Charles River Data Systems, Framingham MA, (617) 626-1101
...!decvax!frog!john, ...!mit-eddie!jfw, jfw%mit-ccc@MIT-XX.ARPA

"Out of my way, I'm a scientist!"
			- War of the Worlds


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