Curvilinear polyhedra as dynamical arenas, illustrated by an example of self-organized locomotion

TL;DR

This paper investigates how dumbbells inside a rotating tilted drum can climb uphill by forming stable pairs, explained through a complex geometric model involving curvilinear polyhedra and unilateral constraints.

Contribution

It introduces a novel geometric framework using curvilinear polyhedra in a high-dimensional space to explain self-organized uphill locomotion.

Findings

Dumbbells form stable pairs that climb uphill against gravity.

Configuration entrapment near polyhedron corners explains stability.

Energetic analysis shows localization near polyhedron corners.

Abstract

Experiment shows that dumbbells, placed inside a tilted hollow cylindrical drum that rotates slowly around its axis, climb uphill by forming dynamically stable pairs, seemingly against the pull of gravity. Analysis of this experiment shows that the dynamics takes place in an underlying space which is a curvilinear polyhedron inside a six dimensional manifold, carved out by unilateral constraints that arise from the non-interpenetrability of the dumbbells. The energetics over this polyhedron localizes the configuration point within the close proximity of a corner of the polyhedron. This results into a strong entrapment, which provides the configuration of the dumbbells with its observed shape that leads to its functionality -- uphill locomotion. The stability of the configuration is a consequence of the strong entrapment in the corner of the polyhedron.