Newton's problem of the body of minimum mean resistance

Abstract

Consider a body Ω at rest in d-dimensional Euclidean space and a homogeneous flow of particles falling on it with unit velocity υ. The particles do not interact and they collide with the body perfectly elastically. Let ℛΩ(υ) be the resistance of the body to the flow. The problem of the body of minimum resistance, which goes back to Newton, consists in the minimization of the quantity (ℛΩ(υ) | υ) over a prescribed class of bodies. Assume that one does not know in advance the direction υ of the flow or that one measures the resistance repeatedly for various directions of υ. Of interest in these cases is the problem of the minimization of the mean value of the resistance ℛ̃(Ω) = ∫ Sd-1 (ℛΩ(υ) | (υ) dυ. This problem is considered (P̃d) in the class of bodies of volume 1 and (P̃dc) in the class of convex bodies of volume 1. The solution of the convex problem P̃dc is the d-dimensional ball. For the non-convex 2-dimensional problem P̃2 the minimum value ℛ̃(Ω ) is found with accuracy 0.61%. The proof of this estimate is carried out with the use of a result related to the Monge problem of mass transfer, which is also solved in this paper. This problem is as follows: find inf T∈script T sign ∫ Π f(ℓ, τ; T(ℓ τ)) dμ(ℓ, τ), where Π = [-π/2, π/2] × [0,1], dμ(ℓ, τ) = cos ℓ dℓ dτ, f(ℓ, τ ℓ′, τ′) = 1 + cos(ℓ + ℓ′), and script T sign is the set of one-to-one maps of Π onto itself preserving the measure μ. Another problem under study is the minimization of ℛ̄(Ω) = ∫ Sd-1 |ℛΩ(υ)| dυ. The solution of the convex problem P̄dc and the estimate for the non-convex 2-dimensional problem P̄2 obtained in this paper are the same as for the problems P̃dc and P̃2.