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New Berry phases in the Foucault pendulum : gyroscopic theory of the spherical pendulum

Verreault René. (2020). New Berry phases in the Foucault pendulum : gyroscopic theory of the spherical pendulum. Proceedings of the Royal Society A - Mathematical, Physical and Engineering Sciences,

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Résumé

In 1984, Michael M. Berry has shown via parallel transport of vectors along a curved hypersurface that the state of a relativistic particle with spin is described not only by the dynamic phase of the cyclic motion, but also by a geometric phase that is totally independent of the spin degree of freedom, depending only on the curved path described by the particle along a geodesic. One year later, John Hannay extended that theory to the case of periodic classical systems described by action-angle variables, such as the geocentric Foucault pendulum. Up to now, nearly plane motion of the pendulum bob parallel to the earth surface has been considered, where a virtual two-form made of two non-degenerate orthogonal circular oscillation states is shown to generate cumulative Berry phases in the form of precession of the oscillation azimuth of the resultant rectilinear oscillation. In this paper, a new kind of two-form made of the two spin states of contrarotating gyroscopes is shown to generate a new set of Berry phases. Since the spin axis is forced to remain horizontal by Earth gravity, the vertical oscillation plane undergoes precession during each half-cycle. Instead of evaluating the effect of the twoform after each complete oscillation cycle, the new two-form is assessed after each half cycle and the difference between the half-cycle effects is cumulated. The new geometric phases are related to the tilt rate of the pendulum vertical, taken as normal to an equipotential hypersurface in free space. Thanks to an 18-hour long pendulum experiment, evidence of a novel 8-shaped orbit is given. The Foucault pendulum is no longer considered in its geocentric environment, but in the barycentric environment of different celestial bodies. Sensitivity to syzygies between pendulum and celestial bodies is discussed.

Type de document:Article publié dans une revue avec comité d'évaluation
Version évaluée par les pairs:Non
Date:2020
Sujets:Sciences naturelles et génie > Sciences naturelles > Astronomie et astrophysique
Sciences naturelles et génie > Sciences naturelles > Physique
Département, module, service et unité de recherche:Départements et modules > Département des sciences fondamentales
Mots-clés:Foucault pendulum, Berry geometric phases, pendulum tilt, syzygies, two-form, gyroscopic effects
Déposé le:12 mai 2020 18:25
Dernière modification:12 mai 2020 18:25
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