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.
Highlights
The motion of the Euler top is governed by the Euler equation, whose generic orbits are periodic
By applying the Stokes theorem to a suitable surface on T ∗SO(3), Montgomery [Mon91] obtained a Berry-Hannay-like formula for the angle by which the final position of the Euler top is rotated with respect to the initial position after one period
The structure of the paper is as follows: the first section briefly reviews the theory of the Euler top; the main result is proved in the second section, with Montgomery’s formula deduced as a corollary in the third section; the fourth section contains an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point
Summary
The motion of the Euler top is governed by the Euler equation, whose generic orbits are periodic. A different derivation, based on the Poinsot description of the motion and the Gauss-Bonnet theorem, was given by Levi [Lev93].1. The purpose of the present paper is to give a third, more elementary derivation, in the sense that it utilizes only basic facts about the Euler equation and parallel transport on the 2-sphere. The basic observation is that the motion of a fixed orthonormal basis as seen in the Euler top’s frame can be understood in terms of the Euler flow on a sphere of fixed angular momentum norm and parallel transport on this sphere. The structure of the paper is as follows: the first section briefly reviews the theory of the Euler top; the main result is proved in the second section, with Montgomery’s formula deduced as a corollary in the third section; the fourth section contains an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point
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