Newton’s problem of minimal resistance under the single-impact assumption

Abstract

A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the body’s surface, the particles are reflected elastically. Assuming that each particle hits the body at most once (the single impact condition (SIC)), the force of resistance of the body along the flow direction can be written down in a simple analytical form. The problem of minimal resistance within this model was first considered by Newton (Newton 1687 Philosophiae Naturalis Principia Mathematica) in the class of bodies with a fixed length M along the flow direction and with a fixed maximum orthogonal cross section Ω, under the additional conditions that the body is convex and rotationally symmetric. Here we solve the problem (first stated in Buttazzo et al 1995 Minimum problems over sets of concave functions and related questions Math. Nachr. 173 71–89) for the wider class of bodies satisfying the SIC and with the additional conditions removed. The scheme of solution is inspired by Besicovitch’s method of solving the Kakeya problem (Besicovitch 1963 The Kakeya problem Am. Math. Mon. 70 697– 706). If Ω is a disc, the decrease of resistance as compared with the original Newton problem is more than twofold; the ratio tends to 2 as → M 0 and to 20.25 as →∞ M . We also prove that the infimum of resistance is 0 for a wider class of bodies with both single and double reflections allowed.