The radial Newton problem: nonlinear dynamics of minimal resistance in central fields

TL;DR

This paper explores the nonlinear dynamics of Newton's minimal resistance problem in radial fields, revealing how physical laws influence the regularity and symmetry of optimal flow configurations.

Contribution

It introduces analysis of non-equilibrium scenarios, demonstrating the regularizing effect of incompressible flow and symmetry-breaking in scale-invariant models.

Findings

Incompressible flow admits unique, smooth, and strictly concave solutions.

Scale-invariant model exhibits symmetry-breaking instability requiring geometric truncation.

Physical conservation laws ensure regularity and symmetry in high-speed central flows.

Abstract

This paper investigates the nonlinear dynamics of Newton's problem of minimal resistance in radial fields. We move beyond classical translational symmetry to analyze two non-equilibrium scenarios: a scale-invariant free expansion and an incompressible source flow. Our analysis reveals that the scale-invariant model suffers from a symmetry-breaking instability (loss of ellipticity) that necessitates geometric truncation. Conversely, we prove that the incompressible flow acts as a structural regularizer, admitting unique, smooth, and strictly concave solutions. These findings provide new qualitative insights into how physical conservation laws ensure the regularity and symmetry of optimal configurations in high-speed central flows, bridging the gap between variational calculus and the physics of complex systems.