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arXiv · 2103.01058

Ants and bracket generating distributions in dimension 5 and 6

Andrei Agrachev·Pawel Nurowski
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Abstract

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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Andrei Agrachev, Pawel Nurowski. 2021-03-01. Ants and bracket generating distributions in dimension 5 and 6. https://doi.org/10.1016/j.automatica.2022.110693

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