for the definition of $K_{p}$ is inserted in order that both $K_{p}$ and $K_{r}$ should be positive for typical rattleback parameters as can be seen below. Note that the asymmetric torque coefficients are dimensionless.

From Eqs. (\[eq:ave-torque\]) and (\[eq:ave-ene\]), $-K_{p}$ is given by $$\begin{aligned}
-K_{p} &= \frac{\braket{\omega_{p}|\,\hat{\varepsilon}(\hat{\Gamma} - 1)\hat{\varepsilon}\hat{\varepsilon}\,|\omega_{p}}}{\braket{ \omega_{p}|\hat{J}|\omega_{p}}}\notag\\
&= -\frac{(a/g)\braket{\tilde{\omega}_{p}|\,\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2}\hat{H}\,|\tilde{\omega}_{p}}}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}\label{eq:kp-1}\\
&= - \omega_{p}^2 \,\frac{(a/g)\braket{\tilde{\omega}_{p}|\,\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2}\,|\tilde{\omega}_{p}}}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}.
\label{eq:kp-2}\end{aligned}$$ In the same way, $K_{r}$ is given by $$\begin{aligned}
K_{r} &= -\frac{(a/g)\braket{\tilde{\omega}_{r}|\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2}\hat{H}|\tilde{\omega}_{r}}}{\braket{\tilde{\omega}_{r}|\tilde{\omega}_{r}}}\label{eq:kr-1}\\
&= \omega_{r}^2 \,\frac{(a/g)\braket{\tilde{\omega}_{p}|(\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{1/2})^{\dagger}|\tilde{\omega}_{p}}}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}.
\label{eq:kr-2}\end{aligned}$$ Equations (\[eq:kp-1\])–(\[eq:kr-2\]) yield simple relations for $K_{p}$ and $K_{r}$ as $$\begin{aligned}
\frac{K_{p}}{K_{r}} =  \frac{\omega_{p}^2}{\omega_{r}^{2}} \label{eq:k-rat}\end{aligned}$$ and $$\begin{aligned}
 K_{p} - K_{r}&= \frac{(a/g)}{\braket{\tilde{\omega}_{p}|\tilde{\omega}_{p}}}
 \mathrm{Tr}\left[\hat{J}^{-1/2}\hat{\varepsilon}\hat{J}^{-1/2}\hat{H}\right]\notag\\
 & = -\frac{1}{2}\sin(2\xi)\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - \frac{1}{\theta}\right). \label{eq:k-diff}\end{aligned}$$ Equations (\[eq:k-rat\]) and (\[eq:k-diff\]) are enough to determine $$\begin{aligned}
    K_{p} = -\frac{1}{2}\sin(2\xi)\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - \frac{1}{\theta}\right) \frac{\omega_{p}^2}{\omega_{p}^2 - \omega_{r}^2},\label{eq:Kp}\\
    K_{r} = -\frac{1}{2}\sin(2\xi)\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)\left(\frac{1}{\phi} - \frac{1}{\theta}\right) \frac{\omega_{r}^2}{\omega_{p}^2 - \omega_{r}^2}\label{eq:Kr}.\end{aligned}$$ Note that Eqs. (\[eq:Kp\]) and (\[eq:Kr\]) are consistent with the three requirements of rattlebacks: $\xi \neq 0$, $\alpha \neq \beta$, and $\theta \neq \phi$. Equations (\[eq:Kp\]) and (\[eq:Kr\]) are shown to be equivalent to the corresponding expressions Eq. (42a,b) in Garcia and Hubbard [@GarciaHubbard1988] although their expressions look quite involved. These results also show that $$K_{p}K_{r} > 0 \quad \text{and hence} \quad \overline{T}_{p}\overline{T}_{r} <0,$$ namely, the torques generated 