蚂蚁和括号生成5维和6维分布
Ants and bracket generating distributions in dimension 5 and 6
摘要：
我们考虑一个三只蚂蚁在地板上的机械系统，它们根据两个独立的规则移动：规则A -迫使任何给定的蚂蚁的速度总是指向相邻的蚂蚁，规则B -迫使每只蚂蚁的速度平行于由其他两只蚂蚁定义的线。我们观察到规则A使蚂蚁的6维构形空间具有齐次（3，6）分布的结构，而规则B将这个6维构形空间叶化成5维叶子，每个叶子都具有齐次（2，3，5）分布。确定了这些分布的对称性和Bryant-Cartan局部不变量。在规则B的情况下，我们研究并确定相应分布的奇异轨迹（异常极值）。我们表明，这些满足一个有趣的系统的Fuchsian型的两个常微分方程。
英文摘要：
We consider a mechanical system of three ants on the floor, which move according to two independt rules: Rule A - forces the velocity of any given ant to always point at a neighboring ant, and Rule B - forces the velocity of every ant to be parallel to the line defined by the two other ants. We observe that Rule A equips the 6-dimensional configuration space of the ants with a structure of a homogeneous (3,6) distribution, and that Rule B foliates this 6-dimensional configuration space onto 5-dimensional leaves, each of which is equiped with a homogeneous (2,3,5) distribution. The symmetry properties and Bryant-Cartan local invariants of these distributions are determined. In the case of Rule B we study and determine the singular trajectories (abnormal extremals) of the corresponding distributions. We show that these satisfy an interesting system of two ODEs of Fuchsian type.
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