Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem
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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 , 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.
KeywordsRigid Body Equilibrium Point Singular Point Saddle Point Convex Body
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