The numerical solution of Newton’s problem of least resistance

Abstract

In this paper we consider Newton's problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in $${\mathbb {R}}^{2}$$ R 2 . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in $${\mathbb {R}}^{1}$$ R 1 . Deriving its Euler---Lagrange equation yields a program with two unknowns, which can be solved quickly.