The robustness of equilibria on convex solids

TL;DR

This paper investigates the minimal perturbations needed to alter the number of static equilibrium points of convex solids, introducing concepts of external and internal robustness, and identifies shapes with maximal robustness in various dimensions.

Contribution

It introduces the concepts of external and internal robustness for convex solids and characterizes shapes with maximal robustness, especially regular polygons and polyhedra, in simplified and full problems.

Findings

Regular polygons have maximal external and internal robustness in 2D.

3D regular polyhedra have maximal internal robustness under certain constraints.

Results suggest why monostatic pebbles are rare in nature.

Abstract

We examine the minimal magnitude of perturbations necessary to change the number $N$ of static equilibrium points of a convex solid $K$. We call the normalized volume of the minimally necessary truncation robustness and we seek shapes with maximal robustness for fixed values of $N$. While the upward robustness (referring to the increase of $N$) of smooth, homogeneous convex solids is known to be zero, little is known about their downward robustness. The difficulty of the latter problem is related to the coupling (via integrals) between the geometry of the hull $\bd K$ and the location of the center of gravity $G$. Here we first investigate two simpler, decoupled problems by examining truncations of $\bd K$ with $G$ fixed, and displacements of $G$ with $\bd K$ fixed, leading to the concept of external \rm and internal \rm robustness, respectively. In dimension 2, we find that for any fixed number $N=2S$, the convex solids with both maximal external and maximal internal robustness are regular $S$-gons. Based on this result we conjecture that regular polygons have maximal downward robustness also in the original, coupled problem. We also show that in the decoupled problems, 3-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in case of 3 dimensional solids. These results appear to explain why monostatic pebbles (with either one stable, or one unstable point of equilibrium) are found so rarely in Nature.