Newton's problem of minimal resistance in Lorentz-Minkowski space

TL;DR

This paper extends Newton's minimal resistance problem to Lorentz-Minkowski space, deriving new equations and solutions, including radial solutions with conical singularities, and analyzing the shock condition.

Contribution

It introduces the Lorentz-Minkowski space version of Newton's problem, deriving the functional, Euler-Lagrange equation, and solutions, highlighting differences from the Euclidean case.

Findings

Euler-Lagrange equation is quasilinear elliptic in Lorentz-Minkowski space.

Radial solutions with conical singularities are characterized.

Maximum principle holds in this Lorentzian context.

Abstract

We extend Newton's problem of minimal resistance to the Lorentz-Minkowski space. We derive the functional energy and determine the Euler-Lagrange equation. In contrast to the Euclidean case, this equation is quasilinear elliptic, and thus, a maximum principle holds in this context. We obtain the solutions of separable variables of this equation via separation of variables. Furthermore, we find all radial solutions to the problem, which present conical singularities at the origin. We also analyze the Single Shock Condition.