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 ‘ $$G_2$$ 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 $$G_2$$ 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 $$G_2$$ G 2 as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $$G_2$$ G 2 geometries is used to construct a ‘ $$G_2$$ G 2 joystick,’ consisting of two balls of radii in ratio $$1\!:\!3$$ 1 : 3 that transforms the difficult $$G_2$$ G 2 manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

中文翻译：

---

飞碟特技

从飞碟是一个非完整机械系统的观察开始，它的 5 维配置空间是一个接触流形，我们展示了如何通过对飞碟的速度施加进一步的非线性限制来用许多几何结构来丰富这个空间。这些限制定义了飞碟的某些“机动”，我们称之为“攻击”、“着陆”或“$$G_2$$G 2 模式”机动，并为其配置空间配备了五种平面抛物线几何形状方面。攻击机动对应于扁平的勒让德接触结构，着陆机动对应于具有特征 (1, 1) 的 Levi 形式的平面超曲面型 CR 结构，最复杂的 $$G_2$$ G 2 机动对应于接触恩格尔结构 (Engel in CR Acad Sci 116:786–788, 1893; Mano et al. 在标记接触扭曲立方结构的几何学，2018 年，arXiv:1809.06455 中）以特殊李群 $$G_2$$G 2 的分裂实数形式作为其对称性。一个著名的关于两个不等价的平面 5 维抛物线 $$G_2$$ G 2 几何形状的双纤维用于构建“$$G_2$$ G 2 操纵杆”，由两个半径为 $$1\! 的球组成： \!3$$ 1 : 3 将困难的 $$G_2$$ G 2 机动转换为飞行员将操纵杆的一个球滚动到另一个球上而不会滑倒或扭曲的动作。