Extremals in the Engel group with a sub-Lorentzian metric

TL;DR

This paper studies extremal trajectories in the Engel group with a Lorentzian metric, proving local maximality of timelike extremals, parametrizing trajectories via Jacobi functions, and analyzing symmetry and cut times.

Contribution

It provides a detailed analysis of extremals in the Engel group with a Lorentzian metric, including parametrization, symmetry group description, and cut time estimates.

Findings

Timelike normal extremals are locally maximizing.

Trajectories are parametrized by Jacobi functions.

A symmetry group and Maxwell points are identified.

Abstract

Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first prove that timelike normal extremals are locally maximizing. Second, we obtain a parametrization of timelike, spacelike, lightlike normal extremal trajectories by Jacobi functions. Third, a discrete symmetry group and its fixed points which are Maxwell points of of timelike and spacelike normal extremals, are described. An estimate for the cut time (the time of loss of optimality) on extremal trajectories is derived on this basis.