An Elementary Derivation of the Montgomery Phase Formula for the Euler Top

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. Introduction 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 (see also [MMR90, Mar92]). A different derivation, based on the Poinsot description of the motion and the Gauss-Bonnet theorem, was given by Levi [Lev93]. 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 easily 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. 1. Euler top In this section we briefly review the theory of the Euler top, mainly to fix the notation. This material is standard and can be found in almost any book on mechanics (e.g. [Arn97, GPS02, MR99, Oli02]). An Euler top is a rigid body with a fixed point moving freely in an inertial frame. Its motion is described by a curve S : R → SO(3) which at each instant gives the orientation of the body with respect to a reference position. We have Partially supported by FCT (Portugal). Both these derivations have been generalized to more complicated mechanical systems (see [AKS95, Cab07]).