Non-holonomic systems as singular limits for rapid oscillations

Mark Levi, Warren Weckesser

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

In this paper, we point out a close relationship between two standard classical problems in mechanics which have coexisted in textbooks for many decades: (1) the pendulum whose suspension point executes fast periodic motion along a given curve; and (2) the skate (known also as the Prytz planimeter, or the 'bicycle'). More generally, we deal with dynamical systems subjected to rapidly oscillating forcing. Examples include: charged particles in rapidly oscillating electromagnetic fields, in particular the Paul trap; particles in an acoustic wave; a bead sliding on a rapidly vibrating hoop. It turns out that the averaged systems of such kind are approximated by a non-holonomic system. The holonomy turns out to have a transparent geometrical or physical interpretation. For the example of a particle in an acoustic wave the holonomy is directly proportional to the speed of the vibration-induced drift.

Original languageEnglish (US)
Pages (from-to)1497-1506
Number of pages10
JournalErgodic Theory and Dynamical Systems
Volume22
Issue number5
DOIs
StatePublished - Oct 1 2002

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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