Geometric phases in the motion of rigid bodies

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Abstract

In this paper we make some differential-geometric observations on the kinematics of convex surfaces rolling along a fixed plane in ℝ3, and on the relationship of the problem with parallel transport and the Gauss-Bonnet formula. These ideas are then applied to recover "Berry's phase" of a free rigid body which was found by Montgomery using Stokes' Theorem. We also point out a new "twist" on this problem. As a second application, we give a solution of a problem posed by R. Brockett. As a third application, we give a geometrical description of "Berry's phase" in SO(3); this can be applied to various rigid-elastic systems to compute their geometric phases.

Original languageEnglish (US)
Pages (from-to)213-229
Number of pages17
JournalArchive for Rational Mechanics and Analysis
Volume122
Issue number3
DOIs
StatePublished - Sep 1 1993

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)
  • Mathematics(all)
  • Mechanics of Materials
  • Computational Mechanics

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