of $n_i <0$ the *steady direction*.

In the small skewness regime, this ratio of the squared frequencies is estimated as $$\begin{aligned}
\frac{\omega_{p}^2}{\omega_{r}^2} \approx \frac{\omega_{p0}^2}{\omega_{r0}^2}
 = \frac{\beta}{\alpha}\frac{1/\phi - 1}{1/\theta -1}. \label{eq:ratomg}\end{aligned}$$ This becomes especially large as $\theta$ approaches $1$ or as $\phi$ approaches 0, namely, as the smaller radius of principal curvature approaches $a$, or as the larger radius of principal curvature becomes much larger than $a$. We remark that both of the inertial parameters $\alpha$ and $\beta$ are larger than $1$ by definition Eq. (\[eq:def-abg\]), and cannot be arbitrarily large for a typical rattleback.

Let us consider these two limiting cases: $\phi\to 0$ and $\theta \to 1$ with $|\xi| \ll 1$. In the case of $\phi \to 0$, $$K_{p} \to \infty,\quad
 K_{r} \to (-\xi) \left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\frac{\alpha}{\beta}\left(\frac{1}{\theta} - 1\right),$$ thus the time for reversal $t_{rGH-}$ remains constant while $t_{rGH+}$ approaches $0$. In the case of $\theta \to 1$, $$K_{p} \to (-\xi) \left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - 1\right),\quad
 K_{r} \to 0,$$ and thus $t_{rGH+}$ remains constant while $t_{rGH-}$ diverges to infinity, i.e., the negative spin rotation never reverses.

Simulation {#sec:simulation}
==========

We perform numerical simulations for the times for the first spin reversal and compare them with Garcia-Hubbard formulas (\[eq:trgh-p\]) and (\[eq:trgh-m\]).

