Hamiltonian

In the present model, we allow the possibility for a global rotation of the warped potential about the $z$-axis. This rotation is parametrised by the angular pattern speed $\Omega_p$ or, equivalently, by the corotating radius $R_c$, taking a negative value for retrograde rotation. Thus, the Hamiltonian in the rotating frame of reference with angular speed $\Omega_p$ is :

$$H=\frac{1}{2}\left( p_x^2 + p_y^2 + p_z^2 \right) +\Phi(x,y,z)-\Omega_p\left( x\,p_y-y\,p_x \right),$$ (5)

where the variables $p_x$, $p_y$, and $p_z$ are the respective canonical momenta of $x$, $y$, and $z$. The equations of motion to integrate are :

$$\begin{array}{l c l c l c l} \dot x&=&p_x + \Omega_p y,& &... ...p_z, & & \dot p_z &=& -\partial_z\Phi_w,\nonumber \end{array}$$

with $\Phi_w$ given by Eq. (3).