Ants and bracket generating distributions in dimension 5 and 6

TL;DR

This paper studies the geometric structures of a mechanical system of three ants on a plane, revealing complex distributions and their invariants, with implications for understanding singular trajectories and abnormal extremals.

Contribution

It characterizes the (3,6) and (2,3,5) distributions arising from ant movement rules, including symmetry properties and Bryant-Cartan invariants, and analyzes singular trajectories.

Findings

Distribution structures are explicitly characterized.

Symmetry properties and invariants are determined.

Singular trajectories satisfy Fuchsian type ODEs.

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.