### 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 language | English (US) |
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Pages (from-to) | 1497-1506 |

Number of pages | 10 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2002 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Ergodic Theory and Dynamical Systems*,

*22*(5), 1497-1506. https://doi.org/10.1017/S0143385702001098