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THE ROBUSTNESS OF EQUILIBRIA ON CONVEX SOLIDS

  • Gábor Domokos (a1) and Zsolt Lángi (a2)

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 $\mathrm{bd} \hspace{0.167em} K$ and the location of the center of gravity $G$ . Here we first investigate two simpler, decoupled problems by examining truncations of $\mathrm{bd} \hspace{0.167em} K$ with $G$ fixed, and displacements of $G$ with $\mathrm{bd} \hspace{0.167em} K$ 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, three-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in the case of three-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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1.Arnold, V. I., Ordinary Differential Equations, 10th printing, MIT Press (Cambridge, MA, 1998).
2.Blaschke, W., Kreis und Kugel, de Gruyer (Berlin, 1956).
3.Bloore, F. J., The shape of pebbles. Math. Geol. 9 (1977), 113122.
4.Dawson, R., Monostatic simplexes. Amer. Math. Monthly 92 (1985), 541546.
5.Dawson, R., Finbow, W. and Mak, P., Monostatic simplexes II. Geom. Dedicata 70 (1998), 209219.
6.Dawson, R. and Finbow, W., What shape is a loaded die? Math. Intelligencer 22 (1999), 3237.
7.Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall (Englewood Cliffs, NJ, 1976).
8.Domokos, G. and Gibbons, G. W., The evolution of pebble shape in space and time. Proc. Roy. Soc. London (2012), doi:10.1098/rspa.2011.0562.
9.Domokos, G., Lángi, Z. and Szabó, T., On the equilibria of finely discretized curves and surfaces. Monatsh. Math. (2012), doi:10.1007/s00605-011-0361-x.
10.Domokos, G., Lángi, Z. and Szabó, T., The genealogy of convex solids. Preprint, 2012, arXiv:1204.5494.
11.Domokos, G., Ruina, A. and Papadopoulos, J., Static equilibria of rigid bodies: is there anything new? J. Elasticity 36 (1) (1994), 5966.
12.Domokos, G., Sipos, A. Á., Szabó, T. and Várkonyi, P., Pebbles, shapes and equilibria. Math. Geosci. 42 (1) (2010), 2947.
13.Domokos, G. and Várkonyi, P., Geometry and self-righting of turtles. Proc. Roy. Soc. London B 275 (1630) (2008), 1117.
14.Firey, W. J., The shape of worn stones. Mathematika 21 (1974), 111.
15.Fejes Tóth, L., Regular Figures, Pergamon (Oxford, 1964).
16.Heath, T. I. (ed.) The Works of Archimedes, Cambridge University Press (Cambridge, 1897).
17.Heppes, A., A double-tipping tetrahedron. SIAM Rev. 9 (1967), 599600.
18.Krapivsky, P. L. and Redner, S., Smoothing a rock by chipping. Phys. Rev. E 9 (2007), 75(3 Pt 1):031119.
19.Krynine, P. D., On the antiquity of sedimentation and hydrology. Bull. Geol. Soc. Amer. 71 (1960), 17211726.
20.Milnor, J., Morse Theory, Princeton University Press (Princeton, NJ, 1963).
21.Poston, T. and Stewart, J., Catastrophe Theory and Its Applications, Pitman (London, 1978).
22.Rayleigh, Lord, Pebbles, natural and artificial. Their shape under various conditions of abrasion. Proc. R. Soc. Lond. A 181 (1942), 107118.
23.Santaló, L. A., Integral Geometry and Geometric Probability, Addison-Wesley (Reading, MA, 1976).
24.Schrott, M. and Odehnal, B., Ortho-circles of Dupin cyclides. J. Geom. Graph. 10 (2006), 7398.
25.Simsek, A., Ozdaglar, A. and Acemoglu, D., Generalized Poincaré–Hopf theorem for compact nonsmooth regions. Math. Oper. Res. 32 (1)193214.
26.Tammes, P. M. L., On the origin of number and arrangement of the places of exit on pollen grains. Diss. Groningen (1930).
27.Várkonyi, P. L. and Domokos, G., Static equilibria of rigid bodies: dice, pebbles and the Poincaré–Hopf theorem. J. Nonlinear Sci. 16 (2006), 255281.
28.Zamfirescu, T., How do convex bodies sit? Mathematica 42 (1995), 179181.
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THE ROBUSTNESS OF EQUILIBRIA ON CONVEX SOLIDS

  • Gábor Domokos (a1) and Zsolt Lángi (a2)

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