Scattering in billiards and problems of Newtonian aerodynamics

Abstract

This paper contains results relating to billiards and their applications to various resistance optimization problems generalizing Newton's aerodynamic problem. The results can be divided into three groups. First, minimum resistance problems for bodies moving translationally in ahighly rarefied medium are considered. It is shown that generically the infimum of the resistance is zero, that is, there are almost 'perfectly streamlined' bodies. Second, arough body is defined and results on characterization of billiard scattering on non-convex and rough bodies are presented. Third, these results are used to reduce some problems on minimum and maximum resistance of moving and slowly rotating bodies to special problems on optimal mass transfer, which are then explicitly solved. In particular, the resistance of a3-dimensional convex body can be at most doubled or at most reduced by3.05% by grooving its surface. Bibliography: 27 titles. © 2009 Russian Academy of Sciences, (DoM) and London Mathematical Society, Turpion Ltd.