Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site utastro.UUCP Path: utzoo!watmath!clyde!burl!ulysses!allegra!bellcore!decvax!genrad!teddy!panda!talcott!harvard!seismo!ut-sally!utastro!anand 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