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 a first glance, one suspects that the two dimensional case should be well known. Nevertheless, the authors have looked into numerous references and asked at least as many experts on the problem, and 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. It is shown in the paper that this is not the case: the two-dimensional problem is richer 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 does not admit a local minimum), it is proven that in dimension two the unrestricted problem is also well-posed when the ratio of height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, it is shown that in the restricted two-dimensional problem the minimizer is not always unique – when the height of the body is less than or equal to its base radius, there exist infinitely many minimizing functions.
