Hubbard [@GarciaHubbard1988], we describe the rattleback dynamics by $\bm{u}$ and $\bm{\omega}$. The evolution of $\bm{\omega}$ is obtained as $$\begin{gathered}
 \hat{I} \dot{\bm{\omega}} -  M\bm{r} \times (\bm{r} \times \dot{\bm{\omega}})
 =  -  \bm{\omega} \times (\hat{I}\bm{\omega}) \\
 + M\bm{r}\times(\dot{\bm{r}}\times \bm{\omega} + \bm{\omega}\times (\bm{r} \times \bm{\omega})) + Mg\bm{r}\times\bm{u} \label{eq:diff-omega}\end{gathered}$$ by eliminating the contact force $\bm{F}$ from the equations of motion (\[eq:em-1\]) and (\[eq:em-2\]), and using the no-slip condition (\[eq:no-slip\]). The state variables $\bm{u}$ and $\bm{\omega}$ can be determined by Eqs. (\[eq:diff-u\]) and (\[eq:diff-omega\]) with the contact point conditions (\[eq:def-z\]) and (\[eq:def-xy\]).

![(color online) \[fig:coordinate\]A body-fixed co-ordinate viewed from below. The dashed lines indicate the principal directions of curvature, rotated by $\xi$ from the principal axes of inertia (the $x$-$y$ axes).](2r.pdf){width="8cm"}

The rattleback is characterized by the inertial parameters $M$, $I_{xx}$, $I_{yy}$, $I_{zz}$, the geometrical parameters $\theta$, $\phi$, $a$, and the skew angle $\xi$. For the stability of the rattleback, both of the dimensionless curvatures $\theta$ and $\phi$ should be smaller than $1$; without loss of generality, we assume $$0 < \phi < \theta < 1,$$ then, it is enough to consider $$-\frac{\pi}{2} < \xi < 0,$$ for the range of the skew angle $\xi$. The positive $\xi$ case can be obtained by the reflection with 