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Gauge fields in the separation of rotations andinternal motions in the n-body problem

Robert G. Littlejohn and Matthias Reinsch

Robert G. Littlejohn and Matthias Reinsch

  • Department of Physics, University of California, Berkeley, California 94720

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Rev. Mod. Phys. 69, 213 – Published 1 January, 1997

DOI: https://doi.org/10.1103/RevModPhys.69.213

Abstract

The problem of separating rotations from internal motions in systems such as macroscopic flexible bodies, atoms, molecules, nuclei, and solar systems is an old one, with many applications in physics, chemistry, and engineering. A new element, however, which has not been appreciated until fairly recently, is the existence of certain gauge fields on the reduced configuration space for such systems. These (non-Abelian) gauge fields arise in the ``falling cat'' problem, in which changes in shape induce changes in external orientation; but they also have a dynamical significance, and enter as gauge potentials in the Lagrangian or Hamiltonian describing the internal or reduced dynamics. Physically these gauge fields represent Coriolis effects. This review concentrates on the case of nonrelativistic, n-body systems not subject to external torques, and develops the gauge theory of rotations and internal motions in detail. Both classical and quantum treatments are given. The gauge theory is developed from the standpoint of classical, coordinate-based tensor analysis; more abstract mathematical notation is generally not used, although the basic geometrical ideas of fiber-bundle theory are developed as needed. Certain old results, such as the Wilson-Howard-Watson Hamiltonian of molecular physics, are examined from a gauge-theoretical standpoint; and several new results are presented, including field equations of the Kaluza-Klein type satisfied by the gauge fields, and geometrical interpretations of the Eckart frame.

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Rev. Mod. Phys. 69, 213– Published 1 January, 1997

    © 1997 American Physical Society

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