the case we will discuss mainly in the following [^1].

Garcia and Hubbard’s theory for the time for reversal {#subsec:GHtheory}
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Based on our formalism, it is quite straightforward to derive Garcia and Hubbard’s formula for the reversal time of rotation.

### Asymmetric torque coefficients

Due to the skewness, the pitching and the rolling are coupled with the spinning motion. We examine this coupling in the case of $\omega_{z} = 0$ by estimating the averaged torques around the vertical axis caused by the pitching and the rolling oscillations. From Eqs. (\[eq:em-1\]) and (\[eq:em-2\]) and the no-slip condition Eq. (\[eq:no-slip\]), the torque around $\bm{u}$ is given by $$\begin{aligned}
 T &\equiv \bm{u}\cdot(\bm{r} \times \bm{F})
 \approx - Ma^2 [\dot{\bm{\omega}}_{\perp}\cdot\hat{\varepsilon}(\hat{\Gamma} - 1)\hat{\varepsilon} \,\bm{u}_{\perp}\,],\label{eq:ave-torque}\end{aligned}$$ within the linear approximation in $\bm{\omega}_{\perp}$, $\bm{u}_{\perp}$, and $\bm{r}_{\perp}$ discussed in Sec. \[subsec:linearization\].

We define the *asymmetric torque coefficients* $K_{p} $ and $K_{r}$ for each mode by $$-K_{p} \equiv \frac{\overline{T}_{p}}{\overline{E}_{p}}, \qquad 
  K_{r} \equiv \frac{\overline{T}_{r}}{\overline{E}_{r}},
  \label{eq:def-kpr}$$ where $\overline{T}_{j}\ (j=p\text{~or~}r)$ is the averaged torque over the oscillation period generated by each mode, and $\overline{E}_{j}$ is the corresponding averaged oscillation energy which can be estimated within the linear approximation as $$\begin{aligned}
 \overline{E}  &\approx Ma^2 (\alpha \overline{\omega_{x}^2} + \beta \overline{\omega_{y}^2} ). \label{eq:ave-ene}\end{aligned}$$ The minus sign 