The robustness of equilibria on convex solids

Domokos, Gábor and Lángi, Zsolt (2014) The robustness of equilibria on convex solids. MATHEMATIKA, 60 (1). pp. 237-256. ISSN 0025-5793

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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 ro- bustness. 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 ex- amining truncations of bd K with G fixed, and displacements of G with bd K fixed, leading to the concept of external and internal 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 maxi- mal 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.

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