{#sec::dynamical-theory}
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I use the usual polar coordinates $\textbf{\textit{x}}=(R,\phi)$ and assume that the temporal evolution of the DF and gravitational potential starts from an axisymmetric equilibrium state described by $f_0(\textbf{\textit{x}},\textbf{\textit{p}})$ and $V_0(R)$ so that $$\begin{aligned}
f(\textbf{\textit{x}},\textbf{\textit{p}},t) &=&
f_0(\textbf{\textit{x}},\textbf{\textit{p}})+
f_1(\textbf{\textit{x}},\textbf{\textit{p}},t), \\
%
V(\textbf{\textit{x}},\textbf{\textit{p}},t) &=&
V_0(R)+V_1(\textbf{\textit{x}},\textbf{\textit{p}},t).
\label{eq:perturbations}\end{aligned}$$ Here $\textbf{\textit{p}}=\left (p_R,p_\phi \right )$ is the momentum vector conjugate to $\textbf{\textit{x}}=\left (R,\phi \right )$. Motion of stars in the equilibrium state is governed by the zeroth order Hamiltonian $${\cal H}_0=\frac 12 \left ( p_R^2+\frac {p_\phi^2}{R^2}\right
)+V_0(R). \label{eq:zeroth-order-H}$$ For bounded orbits, $R$ and $\phi$ become librating and rotating, respectively. One can therefore describe the dynamics using the action variables $\textbf{\textit{J}}=\left (J_R,J_\phi \right )$, $$J_R=\oint p_R  d R,~~J_\phi=\oint p_\phi
d \phi=p_\phi,\label{eq:define-actions}$$ and their conjugate angles $\Theta=(\theta_R,\theta_\phi)$. A transformation $(\textbf{\textit{x}},\textbf{\textit{p}})\rightarrow
(\Theta,\textbf{\textit{J}})$ leaves the Hamiltonian ${\cal H}_0$ as a function of actions only, ${\cal H}_0(\textbf{\textit{J}})$, and therefore, the phase space flows of the equilibrium state lie on a two dimensional torus $\textbf{\textit{J}}=\textbf{\textit{c}}$ with $\textbf{\textit{c}}$ being a constant 2-vector. An action-angle transformation can locally be found for any bounded regular orbit, but it is a global transformation if only one orbit family occupies the phase space. The axisymmetric potential $V_0(R)$ supports only rosette orbits. Radial and circular orbits are 