In previous posts, I indicated my interest in digital simulations of physics that _exactly_ preserved the usual conserved quantities: energy, angular momentum, etc. I hadn't realized that there is a very large field now called "Geometric Integration" that is dedicated to precisely this goal, and that a huge amount of progress has been made since the field took off in approx. 1990. As a result of this progress, it should be possible to import some of the insights from "Geometric Integration" back into the Minsky circle hack paradigm. A good introduction to "Geometric Integrators for ODEs" is by McLachlan and Quispel: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.125.3832&rep=rep1&t... "Six Lectures on the Geometric Integration of ODEs" is also by McLachlan and Quispel: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.130.167&rep=rep1&ty... --- The cool thing about so-called "symplectic" integrators (which preserve "volume" in phase space) is that many of them basically follow the "serialized" structure of the Minsky circle hack model (but without "floor()" or "round()") q_(k+1) = q_k + tau * p_k p_(k+1) = p_k - tau * delV(q_(k+1)) i.e., the second equation depends upon the first, rather than being "simultaneous". This enables the computation to be easily reversed: p_k = p_(k+1) + tau * delV(q_(k+1)) q_k = q_(k+1) - tau * p_k There is also a slightly more symmetrical so-called "leapfrog" version, also with a serialzed structure: q_(k+.5) = q_k + 1/2 * tau * p_k p_(k+1) = p_k - tau * delV(q_(k+.5)) q_(k+1) = q_(k+.5) + 1/2 * tau * p_(k+1) "The total energy is not conserved, but nor does the energy error grow with time, even for time steps of the same order of magnitude as the frequencies of the system. Invariant sets such as periodic, quasiperiodic, and chaotic orbits are well-preserved in phase space. Leapfrog preserves linear and angular momentum (where applicable) up to round-off error. On the other hand, it does become unstable if the time step is too large" There is also an "implicit" symplectic integrator, which should also work for a Minsky-like circle hack in integers: p_(k+1) = p_k + tau*delV((q_k+q_(k+1))/2) q_(k+1) = q_k - tau*(p_k+p_(k+1))/2 Here, we need to solve an implicit equation, possibly by using search or a Newton iteration, in order to find p_(k+1) and q_(k+1). While this "mid-point method" is typically more computationally expensive, the cool thing about it is that: "it too is symplectic, this time for any Hamiltonian system with any constant symplectic or Poisson structure. It preserves any linear symmetries of the system, and is time-reversible with respect to any linear reversing symmetry of phase space. It preserves not just linear and angular momentum but any quadratic first integrals of the system. It is linearly stable for all time steps." The time reversibility comes from the fact that it is no more difficult to solve the implicit equation going backwards than going forwards. The mid-point method does _not_ necessarily conserve energy, either. Indeed: "On the other hand, no method can preserve energy and symplecticity in general." [Ge & Marsden, Phys. Lett. A 133 (1998), 135–139]. --- The orbits of the Minsky "circle" hack obviously don't conserve energy, else the orbits would be exact circles. On the other hand, the closed orbits do have their "energy" bounded both below and above, as well as a "winding number" and an "area". I'm currently doing computer searches to determine if a closed orbit can ever have _retrograde_ motion (= negative area step), how big the winding number can be, and what constraints there may be on the energy bounds of an orbit. For many purposes, the fact that the Minsky circle hack orbits are _disjoint_ is more important than the fact that energy be conserved. This behavior provides a discrete analogy to quantum theory where the different energy "levels" are discrete.