Singularities, collisions and regularizing transformations

An overview of singularities, regularizations and collisions in gravitational N-body systems is presented. The concept of singularity pertains to many fields of science, from the Big Bang theory to black holes, atomic physics, etc. As far as gravitational critical phenomena are concerned, a plethora of collisional events marked the history and evolution of the solar system. The Earth itself experienced many collisions from prehistoric age to recent times due to impacts of asteroids or comets. We report several examples of meteorites and we provide the rate of an impact as a function of the diameter of the colliding object. The standard classification of Near-Earth Objects is presented. From the theoretical point of view, the singularity due to binary collisions between point masses can be handled by means of regularization theory. We review this technique for the limiting case of a two-body system on a line. Coordinate transformations, the introduction of a fictitious time and the conservation of the energy are used to regularize the equations of motion. Triple collisions and the concept of the central manifold are discussed. A simple model, known as the inclined billiard, is presented to investigate chaotic diffusion. Symbolic dynamics is used to characterize the motion, which closely resembles the trajectory of a ring particle. The problem of noncollision singularities is discussed from Painlevé’s conjecture to a 5-body example of noncollision singularities.