Motion of a Rough Disc in Newtonian Aerodynamics

Abstract

Dynamics of a rough disc in a rarefied medium is considered. The main result of the paper is the following: any finite rectifiable curve can be approximated in the Hausdorff metric by trajectories of centers of rough discs (that is, C 0 -small perturbations of regular discs), provided that the parameters of the system are carefully chosen. To control the dynamics of the disc, we use the so-called inverse Magnus effect which causes deviation of the trajectory of a spinning body. We use the shape of the perturbed disc as a parameter to control the Magnus effect. We study the so-called response laws for scattering billiards e.g. relationship between the velocity of incidence and that of reflection. We construct a special family of such laws that is weakly dense in the set of symmetric Borel measures. Then we find a shape of cavities that provides selected law of reflections. Mathematical models for dynamics of rough bodies, corresponding to cavities of the constructed shape, are studied. We write down differential equations that describe motions of such discs. We offer a method on how a given curve can be approximated by considered trajectories.

Billiard in a rotating half-plane: modeling and dynamics

“…Starting from 1993 [5], many mathematical papers studying various settings and approaches in Newton's problem have appeared (see, e.g., [1,4,6,12,13,20,21,22] among others). It is generally assumed in these papers that the body translates in the medium; see, however, the papers [11,19,23,24] where a combination of translational and rotational motions is considered.…”