Extension of a problem of Euler in Lorentz-Minkowski plane

TL;DR

This paper extends Euler's classical problem of moment of inertia to Lorentz-Minkowski space, providing explicit solutions for stationary curves, methods to transform between spacelike and timelike curves, and solving an energy maximization problem.

Contribution

It introduces the Lorentzian version of Euler's problem, derives explicit stationary curve solutions, and develops symmetry-based methods to relate spacelike and timelike curves.

Findings

Explicit solutions for stationary curves in Lorentz-Minkowski space.

A method to transform spacelike curves into timelike ones and vice versa.

Solution to the energy maximization problem for spacelike curves between two points.

Abstract

In this paper we study curves in Lorentz-Minkowski space $\mathbb{L}^2$ that are critical points of the moment of inertia with respect to the origin. This extends a problem posed by Euler in the Lorentzian setting. We obtain explicit solutions for stationary curves in $\mathbb{L}^2$ distinguishing if the curve is spacelike or timelike. We also give a method to carry stationary spacelike curves into stationary timelike curves and vice versa via symmetries and inversions about the lightlike cone. Finally, we solve the problem of maximizing the energy among all spacelike curves joining two given points which are collinear with the origin.