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From: anand@utastro.UUCP (Anand Sivaramakrishnan)
Newsgroups: net.physics
Subject: Re: Re: Re: Non-linear systems.
Message-ID: <1013@utastro.UUCP>
Date: Wed, 16-Jan-85 11:31:05 EST
Article-I.D.: utastro.1013
Posted: Wed Jan 16 11:31:05 1985
Date-Received: Sun, 20-Jan-85 00:55:23 EST
References: <209@talcott.UUCP> <328@rlgvax.UUCP> <384@hou2g.UUCP> <1027@sunybcs.UUCP> <386@hou2g.UUCP> <4781@tektronix.UUCP>
Organization: UTexas Astronomy Dept., Austin, Texas
Lines: 36

>In article <386@hou2g.UUCP> stekas@hou2g.UUCP (J.STEKAS) writes:
>Randomness and predictability are two different things.  Randomness
>is sufficient for unpredictability but not the only source.
>
>In linear types of systems, like Newtonian orbital mechanics,
>one can easily show that intitial states which are infinitesimally
>different at t=0 will be infinitesimally different at t=T.


>>Sorry, but Newtonian mechanics is not necessarily linear. (I'm
>>assuming you're referring to the equation of motion F=ma.)
  .
  .
>>Carl Clawson

I append the following note...

In point of fact, Newtonian gravity is not only non-linear
but also solutions of the equations of motion are very often
'Sensitively Dependent on Initial Conditions' (SDIC).

Frequently higher energy 'orbits' (trajectories) in many
Newtonian systems diverge away from each other in any
neighbourhood through which they pass (i.e. they have
at least one positive 'Liapunov Exponent'). Typical examples
of this phenomenon are found in various models for the
behaviour of (massless) bodies moving under the gravitational
influence of two massive bodies in Keplerian (elliptical
or circular) motion around each other.

This SDIC is so prevalent that it is now a buzzword amongst 
us nonlinear dynamicists. It is the hallmark of 'chaos
in deterministic systems'. This SDIC is found in most
nonlinear differential equations.

				Anand Sivaramakrishnan