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

Varkonyi, PL [Várkonyi, Péter László (alkalmazott mecha...), szerző] Szilárdságtani és Tartószerkezeti Tanszék (BME / ÉPK); Domokos, G [Domokos, Gábor (Nemlineáris mecha...), szerző] Szilárdságtani és Tartószerkezeti Tanszék (BME / ÉPK)

Angol nyelvű Szakcikk (Folyóiratcikk) Tudományos
Megjelent: JOURNAL OF NONLINEAR SCIENCE 0938-8974 1432-1467 16 (3) pp. 255-281 2006
  • SJR Scopus - Engineering (miscellaneous): D1
Azonosítók
Szakterületek:
  • Matematika
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.
Hivatkozás stílusok: IEEEACMAPAChicagoHarvardCSLMásolásNyomtatás
2025-02-12 18:21