Aerobatics of Flying Saucers

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 ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘$G_2$ mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated $G_2$ manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group $G_2$ as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $G_2$ geometries is used to construct a ‘$G_2$ joystick,’ consisting of two balls of radii in ratio $1\!:\!3$ that transforms the difficult $G_2$ manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.