This reminds me of a game we played in high school on graph paper called "racetrack". Very likely we learned about it from a Mathematical Games column. The game board: two simple, non-intersecting closed curves drawn on graph paper that define the sides of a racetrack. The rules: two (or more) players take turns moving their cars from one grid point to another. There are two constraints on the allowed moves. (1) Two cars may not occupy the same grid point and must not stray outside the track. (2) The change in the displacement vector of any player may change, from one move to the next, by at most one unit horizontally or vertically. The first player to finish one lap (without running off the track or crashing into another car) wins. I probably learned more about physics playing this game than I did in physics class. The continuum version suggests some interesting optimization problems that I have never seen worked out. For example, starting at rest on the edge of a circular lake what is the fastest way to run around the lake and arrive, again at rest, at the starting point all the while constraining the acceleration magnitude to be less than 1g? Veit On Jun 10, 2011, at 9:18 PM, Henry Baker wrote:
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?
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