Journal of Nonlinear Science

, Volume 16, Issue 3, pp 255–281 | Cite as

Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem

  • P.L. Varkonyi
  • G. Domokos


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.


Rigid Body Equilibrium Point Singular Point Saddle Point Convex Body 
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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Budapest University of Technology and Economics, Department of Mechanics, Materials and Structures, H-1111 Muegyetem rkp. 1-3, Budapest, Hungary and Center for Applied Mathematics and Computational Physics, BudapestHungary

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