An elementary derivation of the Montgomery phase formula for the Euler top

José Natário

doi:10.3934/jgm.2010.2.113

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2010, Volume 2, Issue 1: 113-118. Doi: 10.3934/jgm.2010.2.113

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An elementary derivation of the Montgomery phase formula for the Euler top

José Natário^1,

1.
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received: August 31, 2009

Revised: January 31, 2010

Published: February 2010

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Abstract

We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the $2$-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.

Keywords:

Euler top,
geometric phase.

Mathematics Subject Classification: Primary 53078.

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