Aerobatics of flying saucers

TL;DR

This paper explores the geometric structures of flying saucers modeled as nonholonomic systems, revealing new flat parabolic geometries and introducing a G2 joystick for maneuver control.

Contribution

It introduces novel geometric structures on the saucer's configuration space by imposing nonlinear velocity restrictions, linking them to flat parabolic geometries and a G2 symmetry framework.

Findings

Identification of three flat parabolic geometries for saucer maneuvers

Development of a G2 joystick model for maneuver control

Connection between geometric structures and control mechanisms

Abstract

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define certain `manoeuvres' of the saucer, which we call `attacking,' `landing,' or `G2 mode' manoeuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manoeuvre corresponds to the flat Legendrean contact structure, the landing manoeuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1,1), and the most complicated G2 manoeuvre corresponds to the contact Engel structure with split real form of the exceptional Lie group G2 as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic G2 geometries is used to construct a `G2 joystick,' consisting of two balls of radii in ratio 1:3 that transforms the difficult G2 manoeuvre into the pilot's action of rolling one of joystick's balls on the other without slipping nor twisting.