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New numerical solutions to Newton's problem of least resistance via a convex hull approach

Gerd Wachsmuth

TL;DR

The paper tackles Newton's problem of least resistance under a convexity constraint in the plane by introducing a convex-hull discretization that models the graph of the minimizer as the upper boundary of a convex hull. It develops two optimization strategies: a free, symmetry-aware approach and a restricted, two-arc framework that exploits observed structural regularities. The restricted method yields highly accurate solutions, resolves extremal arcs, and improves upon prior numerical results, lending support to a two-arc structural conjecture for several height values. Overall, the approach reduces computational complexity, enhances solution accuracy, and points to future work on differential-equation characterizations of the extremal arcs.

Abstract

We present a numerical method for the solution of Newton's problem of least resistance in the class of convex functions using a convex hull approach. We observe that the numerically computed solutions possess some symmetry. Further, their extremal points lie on several curves. By exploiting this conjectured structure, we are able to compute highly accurate solutions to Newton's problem.

New numerical solutions to Newton's problem of least resistance via a convex hull approach

TL;DR

The paper tackles Newton's problem of least resistance under a convexity constraint in the plane by introducing a convex-hull discretization that models the graph of the minimizer as the upper boundary of a convex hull. It develops two optimization strategies: a free, symmetry-aware approach and a restricted, two-arc framework that exploits observed structural regularities. The restricted method yields highly accurate solutions, resolves extremal arcs, and improves upon prior numerical results, lending support to a two-arc structural conjecture for several height values. Overall, the approach reduces computational complexity, enhances solution accuracy, and points to future work on differential-equation characterizations of the extremal arcs.

Abstract

We present a numerical method for the solution of Newton's problem of least resistance in the class of convex functions using a convex hull approach. We observe that the numerically computed solutions possess some symmetry. Further, their extremal points lie on several curves. By exploiting this conjectured structure, we are able to compute highly accurate solutions to Newton's problem.

Paper Structure

This paper contains 5 sections, 11 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Convex hull approach
  3. Free optimization
  4. Restricted optimization
  5. Summary

Figures (2)

  • Figure 1: Numerically obtained solutions of \ref{['eq:free_symmetric']} for the parameters $M = 0.7$, $k = 4$ (top left), $M = 0.9$, $k = 3$ (top right), $M = 1.0$, $k = 3$ (bottom left), $M = 1.5$, $k = 2$ (bottom right).
  • Figure 2: Numerically obtained solutions of \ref{['eq:restricted_symmetry']} for the parameters $M = 0.7$, $k = 4$ (top left), $M = 0.9$, $k = 3$ (top right), $M = 1.0$, $k = 3$ (bottom left), $M = 1.5$, $k = 2$ (bottom right).