The motion of a handle spinning in space has an odd behavior. It seems to unexpectedly flip back and forth in a periodic manner as seen in a popular YouTube video (“Plasma Ben, Dancing T-handle in zero-g, HD,” <https://www.youtube.com/watch?v=1n-HMSCDYtM>). As an asymmetrical top, its motion is completely described by the Euler equations and the equations of motion have been known for more than a century. However, recent concepts of the geometric phase have allowed a new perspective on this classical problem. Here, we explicitly use the equations of motion to find a closed form expression for the total phase and hence the geometric phase of the force-free asymmetric top and we explore some consequences of this formula with the particular example of the spinning handle for demonstration purposes. As one of the simplest dynamical systems, the asymmetric top should be a canonical example to explore the classical analog of the Berry phase.
References
It might be objected that by the time the initial condition ( and ) makes it to the x – z plane, the angular velocity may no longer be of unit magnitude; in fact, in general, it will not). However, since Eq. (20) is independent of , we may renormalize and retain the same Δα, but not necessarily the same exact dynamics. Equivalently (as discussed in Sec. III A), we may rescale our unit of time to give . This requires scaling the time by the ratio of the periods. The Binet ellipsoid and angular momentum sphere are simply magnified or reduced, but the polhode curve shape is maintained.