is the vector from G to the contact point C.

The contact force $\bm{F}$ is determined by the conditions of the contact point; our assumptions are that (1) the rattleback is always in contact at a point with the plane, and (2) there is no slip at the contact point. The second constraint is represented by the relation $$\bm{v} =  \bm{r} \times \bm{\omega}.\label{eq:no-slip}$$ Before formulating the constraint (1), we specify the co-ordinate system. We employ the body-fixed co-ordinate with the origin being the center of mass G, and the axes being the principal axes of inertia; the $z$ axis is the one close to the spinning axis pointing downward, and the $x$ and $y$ axes are taken to be $I_{xx} > I_{yy}$ (Fig. \[fig:coordinate\]). In this co-ordinate, the lower surface function of the rattleback is assumed to be given by $$f(x,y,z) = 0,\label{eq:def-z}$$ where $$f(x,y,z) \equiv \frac{z}{a} - 1 + \frac{1}{2a^2}(x,\,y)\hat{R}(\xi)\hat{\Theta}\hat{R}^{-1}(\xi)\begin{pmatrix}x\\y\\\end{pmatrix},$$ with $$\begin{aligned}
\hat{R}(\xi) \equiv
\begin{pmatrix}
\cos\xi, & -\sin\xi \\
\sin\xi, & \cos\xi \\
\end{pmatrix}, \quad 
\hat{\Theta} \equiv
\begin{pmatrix}
\theta, & 0 \\
0 & \phi \\
\end{pmatrix}.\end{aligned}$$ Here $a$ is the distance between G and the surface at $x=y=0$, and $\xi$ is the *skew angle* by which the principal 