----- Original Message ---- From: Steve Witham <sw@tiac.net> To: math-fun@mailman.xmission.com Sent: Wednesday, April 30, 2008 12:18:23 PM Subject: Re: [math-fun] And I suppose heavy objects fall faster than light ones? ... If everything stopped in its orbit, and was only held up by bouncing against other things (and against the Sun??) it would be like Earth's atmosphere or oceans, or a centrifuge, and then dense stuff would settle toward the center. Then, you've got the fluid pressure pushing up in proportion to size and the gravity pulling down in proportion to mass. I think if the force of gravity dropped off more slowly or more quickly, you'd have the effect of a wall pushing inward or outward, respectively, so this raises a more metaphysical question, why is gravity just so? I wonder whether slight changes in gravity's exponent either way would mean chaotic orbits that collide with each other much more often.   --Steve ______________________________________________ This description, where buoyancy plays a role and denser objects settle lower, appplies to fluids.  For ideal gases, each type of molecule settles to its own Boltzman distribution, independently of the other types of molecules.  In this case, the stratification is according to molecular weight, not molecular density. The 1/r^2 force (and the harmonic oscillator r force) are special in that, for any given orbit, the radial oscillation period equals the angular oscillation period.  If T denotes this common period, we have   r(t + T) = r(t)  and  theta(t + T) = theta(t), so that orbits are closed.  Other force laws produce nonclosed orbits, but slight deviations from Newtonian (e.g. relativistic corrections) produce only slightly nonclosed orbits.  In fact, orbits in the r^1 and r^-2 force fields are related through a z --> z^2 transformation of the complex plane.  This is explained in the very nice book: V. I Arnol'd, "Huygens and Barrow, Newton and Hooke". Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ