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

TL;DR

This paper provides a simple derivation of the Montgomery phase formula for the Euler top's motion, linking geometric phase to holonomy on the 2-sphere, with an interpretation near unstable equilibria.

Contribution

It offers an elementary derivation of the Montgomery phase formula using basic concepts, and provides a geometric interpretation for motions near unstable equilibria.

Findings

Derivation of the Montgomery phase formula using elementary methods

Connection between geometric phase and holonomy on the 2-sphere

Approximate geometric interpretation near unstable equilibrium points

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.