Abstract
By appealing to the Poincare-Hopf Theorem on topological invariants, we introduce a global classification scheme for homogeneous, convex bodies based on the number and type of their equilibria. We show that beyond trivially empty classes all other classes are non-empty in the case of three-dimensional bodies; in particular we prove the existence of a body with just one stable and one unstable equilibrium. In the case of two-dimensional bodies the situation is radically different: the class with one stable and one unstable equilibrium is empty (Domokos, Papadopoulos, Ruina, J. Elasticity 36 [1994], 59-66). We also show that the latter result is equivalent to the classical Four-Vertex Theorem in differential geometry. We illustrate the introduced equivalence classes by various types of dice and statistical experimental results concerning pebbles on the seacoast.
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Varkonyi, P., Domokos, G. Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem. J Nonlinear Sci 16, 255–281 (2006). https://doi.org/10.1007/s00332-005-0691-8
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DOI: https://doi.org/10.1007/s00332-005-0691-8