Henry Baker> All this talk of Gaussian integers reminded me of a problem that I'd dearly love to see solved. Consider N bodies, each with integer masses. Start them in initial positions in a 2D (for the moment, there's plenty of complexity in 2D) which are on integer coordinates. The attractive Newtonian gravitation force is proportional to m*m'/(dx^2+dy^2), so this force is rational. We need to compute new positions for each of the bodies in such a way that all the conservation laws are preserved exactly (if this is possible), yet the number of bits required doesn't grow too quickly (see below). We need to preserve the total energy, we need to preserve momentum & angular momentum, & we might like to preserve information (the system is reversible). Note that we don't necessarily require that delta-t be constant, although if it isn't, we may need to have another counter to represent the "clock" -- i.e., the global time. I don't know how to do this even in the 2-body case, but the Minsky circle hack may provide some guidance -- if we could get the two bodies to each draw a Minsky circle while revolving around one another, then we'd have a good start on the problem. In addition to solving the problem of finite arithmetic, we also need to figure out what happens in collisions. Taking a cue from Fredkin, we should probably have 100% elastic collisions in order to ensure reversibility. How does this square with Poincaré ? Well, I'm assuming that my N-body problem for finite N is Turing complete, so with enough bodies and enough "tape" (empty space), we should be able to compute any computable function. Minsky already provides one potential representation with his "counter machines". So we want a reasonably "efficient" Turing Machine (or Counter Machine) simulation so that we aren't wasting more than a fixed fraction of the bits required to represent the state (N pairs of integers or N gaussian integers). Has anyone attempted this sort of exact simulation before? ------- rwg> The variable speed of the Kepler solution is pretty unMinskylike, although there might be a transformation through which a nonphysical but errorless simulation can appear physical, analogous to the circularization methods in Minskys and Trinskys <http://www.blurb.com/books/2172660>. If http://www.freewebz.com/vitaliy/triApplet/triGrav3.html is any indication, the n=3 case in 2D is unstable with p ~ 1. (Unlike the Windows version, the Apple version seems capable of occasional gross integration errors. But both simulations seem always to blow up.) --rwg --------------- Probably just another false memory, but I thought I publicly fretted over these inevitable blowups. Privately, to the kids, Guys, are you too young to have seen this classic old three-body display hack? http://www.freewebz.com/vitaliy/triApplet/triGrav3.html It always starts with three equal masses, zero net momentum, and, I think, zero net angular momentum about the center, leaving only one degree of freedom in the random in[i]tial conditions. Although the numerical integration is remarkably stable (maybe through "cheating" by nulling out accumulated net momenta and maybe enforcing net energy conservation), it inevitably blows up, despite the existence of at least one known periodic solution. ("Stable" under perturbations O(10^-5).) It is intriguing to try figure out how it (apparently correctly) blows up, ejecting two tightly orbiting particles in one direction and one particle twice as fast in the opposite direction. (It's also interesting to guess which ejections are incomplete, with the particles eventually returning to the center.) The energy for the ejections comes from the tightness of the orbiting pair, which is somehow the result of all three of them nearly colliding, whose probability is enormously enhanced by the 2D constraint. Is the 3D p(eventual ejection) = 1 also? (With a much longer waiting time.) There are known periodic, nonplanar, equal mass three body solutions. Apparently, the relativistic *two* body problem is still incompletely solved! --Bill Duh, it's obvious why the triGrav3 simulations blow up so "unrealistically": Zero angular momentum! Realistic astronomic assemblages come with some "gyro-stabilization" that greatly rarefies the weird encounters that make triGrav3 so visually interesting and unstable. --rwg