# Rattleback dynamics and its reversal time of rotation

## TL;DR

This paper reformulates rattleback dynamics, deriving a simple formula for spin reversal time and exploring its validity and limitations through numerical simulations, revealing complex behaviors beyond the original formula's scope.

## Contribution

It introduces a transparent reformulation of rattleback dynamics and derives a simplified expression for spin reversal time, extending understanding of its behavior.

## Key findings

- The Garcia-Hubbard formula is accurate for small spins and oscillations.
- Fast spins can lead to non-reversal, steady spinning, and chaotic dynamics.
- Numerical simulations reveal limitations of the formula in certain regimes.

## Abstract

A rattleback is a rigid, semi-elliptic toy which exhibits unintuitive behavior; when it is spun in one direction, it soon begins pitching and stops spinning, then it starts to spin in the opposite direction, but in the other direction, it seems to spin just steadily. This puzzling behavior results from the slight misalignment between the principal axes for the inertia and those for the curvature; the misalignment couples the spinning with the pitching and the rolling oscillations. It has been shown that under the no-slip condition and without dissipation the spin can reverse in both directions, and Garcia and Hubbard obtained the formula for the time required for the spin reversal $t_r$ [Proc. R. Soc. Lond. A 418, 165-197 (1988) ]. In this work, we reformulate the rattleback dynamics in a physically transparent way and reduce it to a three-variable dynamics for spinning, pitching, and rolling. We obtain an expression of the Garcia-Hubbard formula for $t_r$ by a simple product of four factors: (1) the misalignment angle, (2) the difference in the inverses of inertia moment for the two oscillations, (3) that in the radii for the two principal curvatures, and (4) the squared frequency of the oscillation. We perform extensive numerical simulations to examine validity and limitation of the formula, and find that (1) the Garcia-Hubbard formula is good for both spinning directions in the small spin and small oscillation regime, but (2) in the fast spin regime especially for the steady direction, the rattleback may not reverse and shows a rich variety of dynamics including steady spinning, spin wobbling, and chaotic behavior reminiscent of chaos in a dissipative system.