Allan is correct, the problem is not well-formulated. There are multiple, potentially conflicting goals. 1. We want a system in 1D, 2D, and/or 3D which preserves the usual conservation & symmetry laws _exactly_ (at least in theory). For computer simulations, rationals are preferred over reals whenever possible, as there is at least the glimmer of hope for exact solutions (perhaps using Chinese Remainder Theorem or other techniques). 2. We're willing to sacrifice continuity for granularity in time & space, so long as the limiting behavior still holds -- e.g., if my space grid points and masses involve integers with 15 decimal digits and relatively slow velocities, then we would hope that the simulation after trillions of iterations is quite close to that of a continuous simulation. 3. We're more interested in long-term & statistical behavior than in short term accuracy/fidelity. We're not trying to land a person on the Moon in 3 days, but to try to get some idea of the Moon's orbit after a billion revolutions. We're also interested in changes in long-term behavior from minor modifications in initial conditions -- e.g., if we change the orbit of Venus by 1 part in 1 million, how does this affect the Earth's orbit in 10 million years? 4. Even if the simulation is significantly different from Newtonian gravity, such a simulation might still be mathematically interesting if one could prove interesting theorems -- e.g., undecidability of long-term stability by reducing the problem to Hilbert's Tenth Problem. 5. If the simulation is sufficient simple, but yields sufficiently interesting behavior, it would be analogous to Conway's "Life" or the 3n+1 problem. At 03:40 PM 6/16/2011, Allan Wechsler wrote:
I have been trying to follow this discussion, and I'm still sort of stuck at the beginning. I'm not sure I understand what Henry is asking for, and I'm not even yet certain that the problem is coherent. Clearly, Henry is look for the definition of a formal dynamical system; furthermore, this system is *not* classical Newtonian gravitational mechanics, since we know the latter will rarely resume rational state once it's set in motion. In other words, the new system might not have anything to say about the actual evolution of a suite of gravitating particles.
The energy, momentum, and angular momentum that Henry wants conserved are, I presume, calculated in the usual way: energy is total gravitational potential plus kinetic energy of the usual mvv/2 sort. Correct me if I'm misunderstanding.
Furthermore, I think we are constraining positions to be integral. The next state is going to have to be selected by the update rule from among all the states with the same E, p, and L. I'm not convinced that there are enough legal next states for the update rule to be at all interesting. Are we convinced that there are *any* other states that are accessible?