Journal of Nonlinear Science

, Volume 16, Issue 3, pp 255–281

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


DOI: 10.1007/s00332-005-0691-8

Cite this article as:
Varkonyi, P. & Domokos, G. J Nonlinear Sci (2006) 16: 255. doi:10.1007/s00332-005-0691-8


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.

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