Static equilibria of rigid bodies: Dice, pebbles, and the Poincare-Hopf Theorem

P. L. Varkonyi, G. Domokos

Research output: Contribution to journalArticle

18 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)255-281
Number of pages27
JournalJournal of Nonlinear Science
Volume16
Issue number3
DOIs
Publication statusPublished - Jun 2006

Fingerprint

Dice
Equivalence classes
Rigid Body
Poincaré
Elasticity
Geometry
Unstable
Theorem
Topological Invariants
Differential Geometry
Convex Body
Equivalence class
Three-dimensional
Experimental Results
Vertex of a graph
Class

Keywords

  • monostatic bodies
  • pebble shapes
  • rigid bodies
  • Static equilibria

ASJC Scopus subject areas

  • Applied Mathematics
  • Modelling and Simulation
  • Engineering(all)

Cite this

Static equilibria of rigid bodies : Dice, pebbles, and the Poincare-Hopf Theorem. / Varkonyi, P. L.; Domokos, G.

In: Journal of Nonlinear Science, Vol. 16, No. 3, 06.2006, p. 255-281.

Research output: Contribution to journalArticle

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