The mechanics of rocking stones: equilibria on separated scales

TL;DR

This paper explains the phenomenon of rocking stones through high-precision scans and Morse-Smale complex analysis, revealing how stable and unstable equilibria form in localized groups called flocks, with implications for understanding their counter-intuitive balancing.

Contribution

It introduces a systematic method using convex hulls and heteroclinic graphs to analyze equilibrium points, demonstrating the typical formation of stable and unstable flocks in rocks and pebbles.

Findings

Equilibria occur in localized groups called flocks.

Most rocks balance on stable local equilibria within stable flocks.

A logarithmic relationship links Zingg parameters to the number of global equilibria.

Abstract

Rocking stones, balanced in counter-intuitive positions have always intrigued geologists. In our paper we explain this phenomenon based on high-precision scans of pebbles which exhibit similar behavior. We construct their convex hull and the heteroclinic graph carrying their equilibrium points. By systematic simplification of the arising Morse-Smale complex in a one-parameter process we show that equilibria occur typically in highly localized groups (flocks), the number of the latter can be reliably observed and determined by hand experiments. Both local and global (micro and macro) equilibria can be either stable or unstable. Most commonly, rocks and pebbles are balanced on stable local equilibria belonging to stable flocks. However, it is possible to balance a convex body on a stable local equilibrium belonging to an unstable flock and this is the intriguing mechanical scenario corresponding to rocking stones. Since outside observers can only reliably perceive flocks, the last described situation will appear counter-intuitive. Comparison of computer experiments to hand experiments reveals that the latter are consistent, i.e. the flocks can be reliably counted and the pebble classification system proposed in our previous work (Domokos et al 2010) is robustly applicable. We also find an interesting logarithmic relationship between the Zingg parameters and the average number of global equilibrium points, indicating a close relationship between the two systems.