Two-dimensional Newton's Problem of Minimal Resistance

TL;DR

This paper explores the two-dimensional Newton's minimal resistance problem, revealing it is more complex and richer than the classical three-dimensional case, with unique and non-unique solutions depending on geometric ratios.

Contribution

The paper demonstrates that the two-dimensional problem is well-posed under certain conditions and uncovers the existence of multiple minimizers in specific cases, unlike the classical three-dimensional problem.

Findings

Two-dimensional problem is well-posed when height-to-radius ratio exceeds a threshold.

In 2D, the minimizer is not always unique, especially when height ≤ radius.

The 2D problem exhibits richer solution structures than the classical 3D case.

Abstract

Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and ask at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two-dimensional problem is more rich than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem doesn't admit a local minimum), we prove that in dimension two the unrestricted problem is also well-posed when the ratio height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted two-dimensional problem the minimizer is not always unique - when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions.