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.

💡 Research Summary

The paper presents a remarkably elementary derivation of the Montgomery phase formula for the Euler top, a classic model of a torque‑free rigid body. The authors start from the well‑known Euler equations,
(I\dot{\omega}+\omega\times(I\omega)=0),
and emphasize two conserved quantities: the magnitude of the angular momentum vector (|\mathbf L|=|I\omega|) and the kinetic energy (E=\tfrac12\omega\cdot(I\omega)). Because (|\mathbf L|) is constant, the tip of (\mathbf L) moves on a sphere (S^{2}) of radius (|\mathbf L|). Moreover, the energy constraint forces the trajectory to lie on a circle of constant latitude on that sphere.

The key geometric observation is that parallel transport of a vector along a closed curve on a unit sphere produces a holonomy equal to the oriented area (\Omega) enclosed by the curve. The authors exploit this fact by interpreting the evolution of the body‑fixed frame as a rotation (R(t)\in\mathrm{SO}(3)) that can be split into a “dynamical” part, generated by the time‑integrated angular velocity component along (\mathbf L), and a “geometric” part, generated by the holonomy on the sphere.

Carrying out the calculation, they obtain
\