A new bizarre algorithm to numerically integrate the classic gravitational N-body problem

As well know, the explicit solution of the classic gravitational N-body problem, does not exist, therefore it is necessary to resort to numerical approximations. The problem is chaotic, anyway if dealing with few bodies and restricting to a time "small" compared to the Lyapunov time, it is almost "deterministic", and, therefore, the right solution can be approached ad libitum by mean of standard integration algorithms for system of ordinary differential equations, both symplectic and not. When aiming to the study of many-body motion over a "long" time compared to that of Lyapunov, what is actually looked for is not an approximation of the "exact" solution, but rather the best possible estimates of the statistical properties of the possible solutions. In such cases and which such aims, the defect of classic algorithms is that of being unnecessarily "precise", expensive, delicate and complex. So, we are proposing a completely different type of algorithm, which in spite of a loss of (unnecessary) precision results to be cheaper, simple and robust in giving reliable evaluation of relevant statistical indicators. The key idea of this type of algorithm is that of replacing the gravitational potential between two bodies, smoothly varying with the inverse of their distance, with a potential, which is kept constant along a certain number of tracts. In such scheme, the gravitational attraction as function of mutual distance assumes a more bizarre behaviour: for almost all distances it is absent except for some distances for which it is infinite. This new problem admits solutions that resemble the "right" solution to arbitrary precision provided an improvement of the approximation of potential, by an increasing refinement of the stepping.