The problem of minimal resistance, old and new

TL;DR

This paper reviews the historical and recent developments in the calculus of variations problem of finding bodies with minimal resistance in fluid flow, highlighting key theoretical advances and proposing new research directions.

Contribution

It provides a comprehensive overview of classical and modern results on minimal resistance shapes and introduces new research directions for future exploration.

Findings

Summary of classical solutions for symmetric bodies

Analysis of convex shape optimization results

Identification of promising future research avenues

Abstract

Since its original formulation by Isaac Newton in 1685, the problem of determining bodies of minimal resistance moving through a fluid has been one of the classical problems in the calculus of variations. Initially posed for cylindrically symmetric bodies, the problem was later extended to general convex shapes, as explored in \cite{BK93}, \cite{BFK95}. Since then, this broader formulation has inspired a number of articles dedicated to the study of the geometric and analytical properties of optimal shapes, with particular attention to their structure, regularity, and behavior under various constraints. In this article, we provide a comprehensive overview of the principal results that have been established, highlighting the main theoretical advancements. Furthermore, we introduce some new directions of research, some of which were described in \cite{P12}, that offer promising perspectives for future investigation.