  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.

Classical Physics