>From David Johnson
> Sorry, if the integration is done with higher precision > it turns out to be the traditional one. > > But it is still useful for determining the gravity from other > geometries. I think it is bad that bodies are approximated with > point sources in their "center of gravity". To the best of my knowledge there is no practical way to map the positions of celestial bodies other than by exploiting brute force differentiation. It's a time-honored methodology, and it works. If you make the iterative slices sufficiently small the results appear to approximate what Nature seems to be doing. Besides, that's what computers are for! Granted, it is conceivable that an integral formula may actually exist, particularly for the simple one body elliptical model. However, to the best of my knowledge, no one has discovered it, or at least published their findings. To be honest, I can't conceive of how integrals could possibly be constructed to predict the characteristics of complicated models, particularly models consisting of two or more bodies. Differentiation alone has enough of a problem dealing with multi-body simulations. Meanwhile, I have been studying what one might consider "the other end" of the spectrum, where I deliberately allow the coarseness of the iterative feed-back loop to manifest in full bloom. I'm not attempting to suppress it. I often welcome their insane effects. With goals of abandon and destruction in mind I've been studying the interesting effects of various Celestial Mechanical computational permutations based on massive iterations of simple algorithms. Some of the results have astonished me. I've discovered the fact that the same simple algorithm can end up producing an astonishing variety of geometric shapes, all with just a simple infinitesimal tweak of a minor constant. The results are often chaotic - and yet surprisingly ordered. One quickly discovers an underlying order that seems to be mysteriously embedded within the chaos. "Ordered chaos" is in fact a curious characteristic of chaos that researchers are finally beginning to realize may hold practical value in possibly help explaining all sorts of subjects such as physics, health, sociology, macro economics - just to name a few. So far, and to the best of my knowledge, no one that I'm aware of seems to have shown much curiosity in analyzing the "life spans" along with the unique characteristics of these algorithms that have been deliberately infected with the virus of chaos. Because of what seems to have been as a "lack of interest" I've come to suspect many researchers may have missed valuable opportunities to analyze an astonishing number of shapes that can be produced from simple iterations of algorithms based on simple k*r, k/r, k/r^2, and k/r^3 - as well as various combinations. Needless to say, over the years I've acquired a much greater respect for the infinite shapes of CHAOS. IMHO, there is precious ore to be mined here. * * * PS: I've just completed a DVD course on Chaos theory. I also plan on taking another DVD course on discrete mathematics, just to make sure I cover as many bases as possible. The DVD courses I'm referring to are produced by an educational company called "THE TEACHING COMPANY". See: http://www.teach12.com/greatcourses.aspx You can find a rich variety of course material available for immediate purchase. All the courses are taught by distinguished professors and teachers. The company often has special sales promotions, where certain courses are reduced from suggested prices of $250 - $300 price tag down to a mere $70. That's how I snapped up several mathematics courses, such as: CHAOS: ($254, marked down to $69) http://www.teach12.com/tgc/courses/course_detail.aspx?cid=1333 DISCRETE MATHEMATICS: ($254, marked down to $69) http://www.teach12.com/tgc/courses/course_detail.aspx?cid=1456 Bon appetite! Regards Steven Vincent Johnson www.OrionWorks.com www.zazzle.com/orionworks
