Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site redwood.UUCP
Path: utzoo!decvax!tektronix!hplabs!hpda!fortune!redwood!rpw3
From: rpw3@redwood.UUCP (Rob Warnock)
Newsgroups: net.physics
Subject: Re: the multi-body problem
Message-ID: <218@redwood.UUCP>
Date: Fri, 27-Sep-85 05:00:24 EDT
Article-I.D.: redwood.218
Posted: Fri Sep 27 05:00:24 1985
Date-Received: Sat, 28-Sep-85 23:58:10 EDT
References: <1330@teddy.UUCP>
Distribution: na
Organization: [Consultant], Foster City, CA
Lines: 95
Xref: tektronix net.physics:03453 

+---------------
| One often hears that the two-body problem (two bodies interacting
| gravitationally) is completely solvable... 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?
+---------------

It has to do with what kinds of differential equations have "closed
form" solutions.  Certain kinds of simultaneous differential equations
are not expressible in "closed form", and this happens to be one of
them, which means: You cannot write a set of equations (a "solution")
for which the left-hand sides are the positions/velocities of the bodies
and the only independent variable on the right-hand side is "time".
(A.k.a.: "Some formulas don't have integrals." Purists: Please excuse
me for the over-simplification.)

The "open form" can, of course, be expressed. I.e., the right-hand
side is a function of time and all of the positions and velocities of
all the bodies, and the left-hand side of each equation is the
derivative of exactly one of those quantities (except time). By
choosing a "small enough" increment of time ("delta_T"), one can
predict "as closely as you choose" what the positions/velocities
are at "T + delta_T", given the positions/velocities at "T". Do
this enough times, and you can predict "as closely as you choose"
the resulting positions/velocities at "T + K*delta_T", for any "K".

The trick comes in knowing:

	a. What "delta_T" do you need to preserve accuracy across
	   "K" iterations?
	
	b. How precise does your arithmetic need to be to preserve
	   accuracy across "K" iterations?
	
	c. How many bodies do you need to consider (i.e., how small
	   a body do you have to include) to get enough accuracy?

+---------------
| 	...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).
+---------------

Right. Unsolvable IN CLOSED FORM has nothing to do with solvable "as
closely as you like" by iterative approximation.  Obviously, NASA and
friends have answered these questions "closely enough".

+---------------
|     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)?
+---------------

Good question. No, there is a special solution for the "2-big/1-small" problem.
Basically you solve the two-body problem, then assume that the third body does
not perturb the "body" comprised of the two-body system (the "2-big" bodies
are considered a single body in solving a two-body problem with the "1-small".
Works fine for (say) spaceship/Earth/Moon. Unfortunately, there are so many
massive bodies in the Solar System that ship/Earth/Moon isn't (practically)
good enough. You need ship/Earth/Moon/Sun/Jupiter/etc.

+---------------
|     3. ...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?
+---------------

Yes. Yes, only worse.

+---------------
|     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. It is NOT SOLVABLE "in closed form", but we do very well
with iterative solutions of the "open form" equations (unknowns on
both sides), thank you.

+---------------
| AN ensuing discussion might be of value, unless the answer is really
| very simple and obvious, which it does not seem to be.  | Dick Pierce
+---------------

Open to constructive flames, I remain...


Rob Warnock
Systems Architecture Consultant

UUCP:	{ihnp4,ucbvax!dual}!fortune!redwood!rpw3
DDD:	(415)572-2607
USPS:	510 Trinidad Lane, Foster City, CA  94404