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Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!mhuxn!charm!grl
From: grl@charm.UUCP (George Lake)
Newsgroups: net.physics
Subject: Re: the multi-body problem
Message-ID: <762@charm.UUCP>
Date: Mon, 30-Sep-85 13:22:24 EDT
Article-I.D.: charm.762
Posted: Mon Sep 30 13:22:24 1985
Date-Received: Thu, 3-Oct-85 04:02:23 EDT
References: <1330@teddy.UUCP>
Organization: Physics Research @ AT&T Bell Labs Murray Hill NJ
Lines: 17

The two-body problem is exceedlingly simple.  You can solve the
equations of both particles by considering their motion in the
center of mass frame.  Conservation of momentume insures that
the center of mass moves at a constant velocity.  Then the
two particles perform symmetric motions about this point.  The entire
motion lies in a plane in this frame.  In some deep sense this occurs
because the "1-body" problem in a central force is so very simple.
Most potentials have 3 distinct invariants of motion.  The central force
has 4, the Kepler problem has 5.  This super-integrability makes
the next higher problem tractable.  At three it goes away.

One way to see it is in terms of resonance.  Two particles orbit one
another with a single frequency.  Three particles have multiple
frequencies. When the frequencies are equal, resonance makes the motion
wild and difficult to calculate.  There are infinitely many resonances
that all have be calculated.  Each resonance is a singularity and
perturbation calculations won't go through them.


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