in §\[sec:modes-exp-disk\].

Equating (\[eq:expansion-V1\]) and (\[eq:expansion-V1-config\]), multiplying both sides of the resulting equation by $\exp[-{\rm i}(l\theta_R+m\theta_{\phi})]$ and integrating over the $\left (\theta_R,\theta_{\phi} \right )$-space, lead to (see also Kalnajs 1977 and Tremaine & Weinberg 1984) $$\begin{aligned}
&{}& \!\!\! \sum_{j=0}^{\infty} b^{ml}_j(t) \Psi^{ml}_j(\textbf{\textit{J}})
  \!=\! \sum_{j=0}^{\infty} a^m_j(t) \tilde \Psi^{ml}_j(\textbf{\textit{J}}),
 \label{eq:relation-b-and-a} \\
%
&{}& \!\!\! \tilde \Psi^{ml}_j(\textbf{\textit{J}}) \!=\!
     {1\over \pi}\int\limits_{0}^{\pi}
\psi^{|m|}_j(R) \cos [l\theta_R+m(\theta_{\phi}-\phi)] d \theta _R,
 \label{eq:fourier-coeffs}\end{aligned}$$ where $\tilde \Psi^{ml}_j$ are the Fourier coefficients of the basis potential functions in the configuration space. The trial functions used in the expansion of $V_1$ in the action-angle space are not necessarily identical to $\tilde \Psi^{ml}_j$. However, subsequent mathematical derivations are greatly simplified by setting $\Psi^{ml}_j(\textbf{\textit{J}})= \tilde
\Psi^{ml}_j(\textbf{\textit{J}})$, which implies $b^{ml}_j(t)=a^m_j(t)$. To build a relation between $d^{ml}_j(t)$ and $a^m_j(t)$, I use the fundamental equation $$f_1(\Theta,\textbf{\textit{J}},t) d \textbf{\textit{J}} d \Theta
=\Sigma_1(R,\phi,t) R dR d \phi,
\label{eq:fundamental-equation}$$ where $ d \textbf{\textit{J}} d \Theta$ is the volume of an infinitesimal phase space element. On substituting (\[eq:expansion-f1\]) and (\[eq:expansion-sigma1-config\]) in (\[eq:fundamental-equation\]), multiplying both sides of the resulting equation by $\psi^{|m|}_j(R)e^{-{\rm i}m\phi}$ and integrating, one obtains $$\begin{aligned}
a^m_j(t)&=& \frac
{4\pi^2}{D_j(m)}\sum_{l=-\infty}^{\infty}\sum_{p=0}^{\infty}
\Lambda^{ml}_{jp}d^{ml}_p(t),\label{eq:a-versus-d} \\
%
\Lambda^{ml}_{jp} &=& \int
\Psi^{ml}_j(\textbf{\textit{J}})
\Phi^{ml}_p(\textbf{\textit{J}}) d \textbf{\textit{J}},\end{aligned}$$ which is inserted in (\[eq:expansion-V1\]) to represent $V_1$ in terms of the 